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December 25, 2010

An Informal Introduction to Topos Theory

Posted by Tom Leinster

Christmas is a time for giving, but it is also a time for topos theory. (At least, it is in my own private tradition; I don’t know about you.) Combining the two, I give you:

An informal introduction to topos theory

This came out of some impromptu talks I gave to a bunch of category theorists earlier in the year. In odd moments since then I’ve been typing up notes. Voilà!

I’ll be glad to hear your comments.

I’ll also be glad if anyone can suggest somewhere this might be publishable. While ideally I’d like to get it into a journal, I suspect it will probably just end up on the arXiv, because it’s purely expository and doesn’t really adopt a novel point of view. Its only possible claim to novelty is that it goes lightly over quite a lot of ground in not many pages. I had a look at the question about expository articles at MathOverflow, but no journal listed there seemed clearly appropriate. Any ideas?

Posted at December 25, 2010 12:16 AM UTC

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Re: An Informal Introduction to Topos Theory

Thanks! Topos theory is not in my personal Christmas tradition, but expository LaTeXing is. As it turns out, the only Christmas present I have given this year is an informal and (I hope) approachably brief introduction to operads, which I finished this morning.

The only Christmas present I have received is now an informal introduction to topos theory - how pleasingly symmetric.

By the way, before anyone tries to pity me about the presents, let me say that I read somewhere that approximately £2 billion was spent on unwanted Christmas presents just in the UK last year. I decided to remove myself from that equation several years ago.

Merry, or suicidal, Christmas, depending on your inclinations!

Posted by: Eugenia Cheng on December 25, 2010 8:13 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

John ends his TWF week 190 with a remark on operads in the context of “categorification of C”. It would be great if one could read more about that (or have I just missed where to look?).

Posted by: Thomas on December 25, 2010 8:28 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’ll say Merry Christmas, Eugenia!

Some years back I made a pact with my immediate family: no presents between the adults. (There’s really no getting around giving presents to the children.) I do exchange gifts with my wife, but my parents and my brother and I are very happy not having to deal with buying gifts for each other.

As long as we’re on the topic of the annoyances with Christmas, I’ll mention the constant barrage of Christmas songs one hears while going about one’s business. (Is that a US thing?) Some of the traditional carols are lovely, but the majority of songs are just plain grating! (I really can’t stand “It’s the most wonderful time of the year”!)

I like the expository gift idea; hadn’t heard of that one.

Posted by: Todd Trimble on December 25, 2010 10:13 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’ve read it now - it’s lovely, just what I’ve always wanted for Christmas deep down ever since I stopped wanting anything for Christmas. Now I know what to ask for next Christmas, if I actually have to suffer another Christmas.

Posted by: Eugenia Cheng on December 25, 2010 9:16 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Why thank you!

The world wants to know (and I speak on its behalf): how can we get your introduction to operads?

Posted by: Tom Leinster on December 26, 2010 6:11 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Being a non-mathematician who got stuck on Toposes Triples and Theories, and might go back to it when I’ve finished Categories for the Working Mathematician, I thought it was rather good.

The first chapter I’d heard before, the second was nice (is it possible to do a ‘toposes and set theory’ that isn’t?) and I learned stuff too. The third was a bit over my head (geometry and topology almost invariably are), but I’ve wondered what a geometric morphism is for a while. The final chapter looked like a nice exposition of what I hope to learn formally soon.

As a quick aside: you didn’t give the full expansion of TQFT or I think PROP. Not that that’s necessarily a bad thing: I don’t know what they mean, so can’t judge.

Posted by: Jon Phillips on December 27, 2010 2:20 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks for the comments, Jon. TQFT is “topological quantum field theory”; I’ll probably expand that. I think PROP stands for “PROducts and Permutations”, or something like that, but it’s one of those words like “radar” where the unabbreviated form has been mostly forgotten. So I think I’ll leave that one as it is.

Posted by: Tom Leinster on December 27, 2010 7:10 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

stands for “PROducts and Permutations”

That’s correct.

Posted by: Todd Trimble on December 27, 2010 8:00 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

On the bottom of page 16 it says:

I do not know a short way to explain why the subtoposes of ^\hat \mathbb{C} correspond to the Grothendieck topologies on \mathbb{C}.

A possibly useful intermediate stepping stone is a general fact about reflective subcategories: every reflective subcategory CLDC \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} D of a (locally presentable) category DD is the full subcategory on the L 1(Isos)L^{-1}(Isos)-local objects: those objects XDX \in D such that D(f,X)D(f,X) is an iso if L(f)L(f) is.

Using that in a presheaf category DD every object is a colimit of representables, one finds that in this case it is sufficient to look at those f:SUf : S \to U with representable codomain UU. These are the “covering sieves” of UU. The condition that D(f,X)D(f,X) is an iso is then the sheaf condition on the presheaf XX.

Finally, the requirement that LL is left exact can be seen to correspond to pullback stability of these covering sieves.

Posted by: Urs Schreiber on December 28, 2010 4:21 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I don’t understand why it suffices to restrict to representable UU. An extension of SXS \rightarrow X along ff is determined by a cone of maps from the representables in the colimit diagram for UU, but I don’t see how this helps.

A related question, how could I have copied the relevant paragraph from the above post into my comment? Is the procedure to TeX it by hand and then use the html “blockquote” tag?

Posted by: Emily Riehl on January 1, 2011 1:02 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I wrote:

Using that in a presheaf category DD every object is a colimit of representables, one finds that in this case it is sufficient [for the description of reflective localizaitons/sheafification] to look at those f:SUf : S\to U with representable codomain UU.

Emily Riehl said:

I don’t understand why it suffices to restrict to representable UU.

You can find the full proof written out on the nnLab at category of sheaves (there done for \infty-categories, but it is verbatim the same as for 1-categories) in the section Localization and Grothendieck topology

The key argument that I mentioned is in more detail of the following type:

for f:YXf : Y \to X any morphism, write the codomain as a colimit of representables

Xlim iU i X \simeq {\lim_\to}_i U_i

over some diagram II. Then consider the pullback diagram

f *(lim iU i) Y f lim iU i X. \array{ f^*({\lim_\to}_i U_i) &\to& Y \\ \downarrow && \downarrow^{\mathrm{f}} \\ {\lim_\to}_i U_i &\stackrel{\simeq}{\to}& X } \,.

Since we have universal colimits in the presheaf topos, this is equivalently

(lim if *U i) Y f lim iU i X. \array{ ({\lim_\to}_i f^* U_i) &\to& Y \\ \downarrow && \downarrow^{\mathrm{f}} \\ {\lim_\to}_i U_i &\stackrel{\simeq}{\to}& X } \,.

But since this is a pullback along an equivalence, it follows that also

(lim if *U i)Y. ({\lim_\to}_i f^* U_i) \stackrel{\simeq}{\to} Y \,.

This means that the above diagram realizes the morphism f:YXf : Y \to X as a colimit over morphisms with representable codomain.

Now, it is clear that if a reflective localization LL

ShLPSh Sh \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh

inverts a set S 0S_0 of morphisms, then it inverts in particular also all colimits over elements in S 0S_0 (because LL preserves colimits so that they are sent to colimits of equivalences which are equivalences).

This shows that for specifying a reflective localization of a presheaf topos, it is sufficient to specify the morphisms with representable codomain that are being inverted.

A further argument shows that moreover it is sufficient to look at monomorphisms with representable codomain that are being inverted, the dense monomorphisms with representable codomain. These are the covering sieves.

(This last step however is a little different in the \infty-case, where one has to distinguish between topological localizations at monomorphisms on one end and hypercompletions on the other.)

Posted by: Urs Schreiber on January 1, 2011 11:40 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Meant to write much earlier to say thanks.

Posted by: Emily Riehl on January 13, 2011 5:11 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

how could I have copied the relevant paragraph from the above post into my comment?

There’s no way to copy a paragraph with math in it unless the person writing the paragraph was considerate enough to sign their comment with PGP. If they did that, then you can click on “PGP Sig” and the blog software will also show you the itex source of the page. For instance, I’ve signed this comment, and (h er) eis𝕄αθ(h^{er})_e \; \mathbf{is} \; \sum\; \mathbb{M}\alpha\theta.

I think usually what people do when quoting text with math is to manually re-itex the math; at least, that’s what I do. But you don’t need to use html tags, if you choose a “Markdown with itex2mml” text filter you can just put a “>” in front of the text you copied and it will get blockquoted automatically.

Posted by: Mike Shulman on January 1, 2011 6:54 AM | Permalink | PGP Sig | Reply to this

Re: An Informal Introduction to Topos Theory

𝕄αθ\sum\; \mathbb{M}\alpha\theta.

Cute. And thanks.

Posted by: Emily Riehl on January 13, 2011 5:14 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Here’s something I currently understand at a mechanistic level, but would like to understand at a conceptual level.

It’s about subtoposes and Lawvere-Tierney topologies. The theorem mentioned in my notes is that given a topos EE, subtoposes of EE correspond one-to-one with Lawvere-Tierney topologies on EE.

A subtopos of EE is a subcategory of EE with certain properties.

A Lawvere-Tierney topology on EE is a subobject of Ω E\Omega_E with certain properties. (It’s usually presented as a map Ω EΩ E\Omega_E \to \Omega_E with certain properties, but that’s the same thing.)

So:

a subcategory of EE with certain properties

is the same as

a subobject of Ω E\Omega_E with certain properties.

Is there a nice story explaining this?

Posted by: Tom Leinster on January 2, 2011 9:37 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

if i’m not too badly confused then you’re suggesting an analogy between the sheaves for a lawvere-tierney topology j and the “j-dense truth values”. that seems screwed up; instead, try using the “j-closed truth values”, which should give a much better analogy.

the j-dense truth values are the ones that get promoted to “true” by j, forming a sort of kernel if you think of “true” as zero. the j-closed ones are in contrast the ones that get promoted to themselves, so the only truth value that’s both j-dense and j-closed is “true” itself. so j-dense and j-closed are rather antagonistic concepts, and getting them confused with each other could screw up the intuition a lot.

Posted by: james dolan on January 2, 2011 7:40 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks, Jim. Could you help me out a bit more here?

I understand the definitions of jj-dense and jj-closed truth value. But why do you say that I’m suggesting an analogy between sheaves and jj-dense truth values? (I’m not saying that I’m not; I’m simply in the dark.)

And when you say that using jj-closed truth values should give a much better analogy, what analogy do you have in mind?

Thanks.

Posted by: Tom Leinster on January 2, 2011 8:26 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

well, i probably should have said that you were hinting at some sort of parallelism rather than explicitly suggesting an analogy, just from the way that you wrote “a subcategory of e with certain properties” parallel to “a subobject of [the truth-values object of e] with certain properties”.

the analogy that i have in mind that does work is, roughly, that a global point p of the truth-values object in a topos corresponds to an object x_p whose morphism to the terminal object is monic (aka “subobject of 1”); and that under that correspondence, p is j-closed iff x_p is a j-sheaf. (there’s probably a better way of expressing that though; perhaps “j is itself the j-sheafification process for subterminal objects”.)

anyway, since you mentioned that it’s possible to encode a lawvere-tierney topology j as the subobject classified by j (“the j-dense truth-values”), i just wanted to mention that you can instead encode it as the subobject given as the image of j (“the j-closed truth-values”), and that this might be more useful than the j-dense truth-values in developing the kind of story that you’re asking for.

Posted by: james dolan on January 2, 2011 11:03 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

some of the analogy that i mentioned works more straightforwardly in the case of a localic topos; then the need to treat the truth-values “internally” rather than “externally” is lessened because there are enough global truth-values to characterize the situation.

Posted by: james dolan on January 3, 2011 5:27 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks, Jim. I understand better what you mean now.

Posted by: Tom Leinster on January 4, 2011 9:35 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I think what Jim means is that when you said

A Lawvere-Tierney topology on EE is a subobject of Ω E\Omega_E with certain properties. (It’s usually presented as a map Ω EΩ E\Omega_E\to\Omega_E with certain properties, but that’s the same thing.)

it sounds like you mean to look at the subobject classified by the map j:Ω EΩ Ej\colon \Omega_E\to\Omega_E (using the fact that Ω E\Omega_E is the subobject classifier). If so, then that gives you the “object of j-dense truth values,” since the way you get the subobject classified by jj is to look at the inverse image of Ω E\top\in\Omega_E, and jj is (internally) the map which takes the closure of a subobject. Thus, j 1()j^{-1}(\top) is the object of truth values whose closure is \top, i.e. which are dense.

There is a different subobject of Ω E\Omega_E consisting of the jj-closed truth values: it’s the splitting of the idempotent jj, or its “object of fixed points”. Better yet, it’s the object of jj-algebras, since jj is an idempotent monad on the internal poset Ω E\Omega_E.

This brings us to the analogy to sheaves. One way to describe it is to consider the analogue of jj to be the left-exact reflector L:EEL\colon E\to E. The corresponding subcategory of EE is the “fixed subcategory” of LL, or the category of LL-algebras, which is the subcategory into which LL reflects.

Thus if we take the slogan to be subtoposes are left-exact reflective subcategories (= left-exact idempotent monads), and observe that Ω E\Omega_E is (the frame corresponding to) an internal locale, i.e. an internal (0,1)-topos in EE (namely, the terminal locale in EE), then the internal left-exact idempotent monad jj is the “internalization” of the “external” left-exact idempotent monad LL.

Posted by: Mike Shulman on January 2, 2011 10:58 PM | Permalink | PGP Sig | Reply to this

Re: An Informal Introduction to Topos Theory

Another part of the story is the following (probably there are some details that make this not entirely true but it is at least morally true): Ω E\Omega_E is an internal monoidal category in EE and the internal Cauchy-complete categories enriched over Ω E\Omega_E are just objects of EE. The object Ω j\Omega_j of jj-closed truth values is also an internal monoidal category and the internal Cauchy-complete categories enriched over Ω j\Omega_j are jj-sheaves. There is an internal monoidal adjunction between Ω E\Omega_E and Ω j\Omega_j and this induces by change of base the inclusion/localization adjunction between EE and the jj-sheaves in EE.

Posted by: Richard Garner on January 4, 2011 4:35 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

OK, thanks. Now I get a clearer picture of jj-dense vs. jj-closed.

Posted by: Tom Leinster on January 4, 2011 9:40 PM | Permalink | Reply to this

big and little toposes

Section 3 summarizes the relation of toposes to geometry as

a topos is a generalized space

Strictly speaking, this misses an aspect of topos theory in geometry: not every topos is like a generalized space, some are like categories of generalized spaces .

Of course the distinction depends on what one counts as a generalized space. Often it is not made. But that one should make a distinction goes back to remarks by Grothendieck and has been emphasized over many years by Bill Lawvere, first in

Bill Lawvere, Categories of spaces may not be generalized spaces .

The two kinds of toposes in geometry are sometimes called gros and petit toposes or big and little toposes. A little topos is to be regarded as a generalizeed space itself, a big topos is to be regarded as a category of generalized spaces.

What Lawvere has been proposing ever since the above note is an axiomatics for big toposes . The solution is quite beautiful: a Grothendieck topos \mathcal{E} is a big topos of generalized spaces – Lawvere introduced for this the nicely descriptive term topos of cohesion – if the terminal geometric morphism (ΔΓ):Set(\Delta \dashv \Gamma) : \mathcal{E} \to Set to the base has an extra left adjoint and an extra right adjoint to a total of a quadruple of adjoint functors

(Π 0DiscΓCodisc):Set (\Pi_0 \dashv Disc \dashv \Gamma \dashv Codisc) : \mathcal{E} \to Set

such that Π 0\Pi_0 preserves finite products. This has the following meaning in terms of the interpretation of objects of \mathcal{E} as generalized spaces:

  1. the underlying set of points of an object XX \in \mathcal{E} is Γ(E)\Gamma(E);

  2. the set of connected components of points of XX is Π 0(X)\Pi_0(X).

This allows us to think of XX \in \mathcal{E} as being a set of points that are lumped together by some kind of cohesion . For instance by topology, or by smooth structure.

Then

  1. For SS a set, Disc(S)Disc(S) is the discrete cohesion on this set (for instance the discrete topology on SS, or the discrete smooth structure);

  2. For SS a set, Codisc(S)Codisc(S) is the codiscrete cohesion on this set (for instance the codiscrete topology on SS, or the codiscrete smooth structure).

For instance some smooth toposes modelling the axioms of synthetic differential geometry are cohesive, reflecting the fact that they are categories of generalized smooth spaces (for instance the Cahiers topos).

In his original note Lawvere observes that a topos that satisfies the axioms of a cohesive topos is genuinely big in that it cannot be a localic topos. The meaning of this again depends a bit on which standpoint one takes towards generalized spaces. One can make this a positive statement and observe that as generalized spaces, cohesive toposes are fat points , namely the abstract cohesive lump of points on wich the given notion of generalized space is modeled.

In the Idea-section of the nnLab entry topos we have some general remarks on the usefulness of the two perspectives of big and small toposes in geometry. For instance for every object XX in a big/cohesive topos \mathcal{E}, the over-topos /X\mathcal{E}/X is the little topos incarnation of XX, thus extracting the generalized space XX from inside the big topos to a stand-alone little topos.

This and various other aspects of cohesive toposes becomes even clearer and more pronounced as we pass from toposes to (,1)(\infty,1)-toposes – as we should, lest we are being evil and impose equations where there are only equivalences.

The above definition immediately generalizes to give the notion of cohesive (,1)(\infty,1)-toposes. It is quite remarkable how much property and structure is implied in and on a cohesive (,1)(\infty,1)-topos, discussed in detail at the above link.

Notably there is a nice relation to classifying toposes for essentially algebraic theories. Thiss closes a grand circle and shows how this all fits together into one grand general abstract picture of geometry as such (all \infty-s implicit from now on):

A category 𝒢\mathcal{G} that both has finite limits and the structure of a site in a compatible way we call a geometry. Examples to keep in mind are

  • formal duals to finitely presented rings with the é topology: this models algebraic geometry;

  • smooth manifolds with the open cover topology: this models differential geoemtry.

Then the sheaf topos

H=Sh(𝒢) \mathbf{H} = Sh(\mathcal{G})

is the category of generalized spaces modeled on the geometry 𝒢\mathcal{G}. At least in the case of 𝒢=Manifolds\mathcal{G} = Manifolds this is cohesive.

On the other hand, let 𝒳\mathcal{X} be a little sheaf topos. Then a finite-limit preserving functor

𝒪 𝒳:𝒢𝒳 \mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}

is a 𝒢\mathcal{G}-algebra in 𝒳\mathcal{X}: a 𝒢\mathcal{G}-valued structure sheaf. If this preserves coverings then this makes (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) a locally ringed topos with respect to the notion of “ring”=algebra encoded by the essentially algebraic theory 𝒢\mathcal{G}.

Every object XHX \in \mathbf{H} ought to be canonically a locally 𝒢\mathcal{G}-ringed topos, and it is: the little over-topos H/X\mathbf{H}/X sits canonically over H\mathbf{H} by an étale geometric morphism

H/XX X X !H \mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathbf{H}

and composed with the Yoneda embedding jj this gives the canonical structure sheaf (= algebra over the theory 𝒢\mathcal{G})

𝒪 X:𝒢jHX *H/X. \mathcal{O}_X : \mathcal{G} \stackrel{j}{\to} \mathbf{H} \stackrel{X^*}{\to} \mathbf{H}/X \,.

So we see that the two or three roles of toposes as big or little toposes of generalized spaces and as classifying toposes of essentially algebraic theories are really all different aspects of one single general abstract concept: geometry.

  1. Either one probes a generalized space by mapping test spaces in 𝒢\mathcal{G} into it: this gives the notion of a sheaf over 𝒢\mathcal{G};

  2. or one co-probes a generalized space by mapping out of it into test spaces in 𝒢\mathcal{G}: this gives the 𝒢\mathcal{G}-valued structure sheaf on the space and hence an algebra over 𝒢\mathcal{G}.

The unification of geometry and algebra at work here is essentially what is known as Isbell duality. Topos theory, with its duality of big versus little toposes, of cohesive versus locally ringed toposes, classifying toposes versus localic toposes is in a nice realization of this fundamental duality: a space may be detected either by mapping probes into it or by mapping out of it into probe objects. Lawvere called this fundamental duality that of space and quantity . Geometry and algebra. Two sides of the same coin.

Posted by: Urs Schreiber on December 29, 2010 2:13 AM | Permalink | Reply to this

Re: big and little toposes

Thanks for this. The question of which toposes really deserve to be thought of as generalized spaces is interesting, I think.

I wonder how one decides what constitutes a good answer to that question. Of course, one can always employ one’s mathematical nose: ‘this smells right’. But I don’t know what criteria are used by people who work seriously on these things.

I guess you know this, but just to be clear: I meant the slogan ‘a topos is a generalized space’ in its most common, primitive sense—the class of (sober topological) spaces embeds into the class of toposes. In the same way, if I was explaining the definition of monoid to someone who knew what groups were, I might tell them that monoids were a generalization of groups. Later, if they get interested in monoids, they’ll discover that some monoids behave a lot more like groups than others.

(As you’ll have guessed, I’m not going to put anything about cohesion and big/little toposes into the article. I’ve already failed to resist the irresistible temptation to write more than I actually said in the talks.)

I’ve been following the unfolding axiomatization of cohesive toposes from a distance (i.e. interested but not wholly paying attention). Let me see if I can understand something simple.

Take a small category C\mathbf{C} with both an initial and a terminal object. Then according to the definition in your comment, Psh(C)=Set C opPsh(\mathbf{C}) = Set^{\mathbf{C}^{op}} is a topos of cohesion (== cohesive topos?).

Proof:  the unique functor C1\mathbf{C} \to \mathbf{1} has both a left and a right adjoint. This chain of three adjoint functors between C\mathbf{C} and 1\mathbf{1} gives rise, by Kan extension, to a chain of five adjoint functors between Psh(C)Psh(\mathbf{C}) and Psh(1)=SetPsh(\mathbf{1}) = Set. Throw away the leftmost functor. What remains is a chain of four adjoint functors (going in the correct direction), with the leftmost one preserving all limits, and in particular finite products. QED.

On the other hand, it’s not true for arbitrary small categories C\mathbf{C} that Psh(C)Psh(\mathbf{C}) is a topos of cohesion. Maybe someone knows exactly which C\mathbf{C} it is true for. (I couldn’t see it on the nnLab page.)

My questions: why does Psh(C)Psh(\mathbf{C}) deserve to be thought of as a category of generalized spaces for any small category C\mathbf{C} with initial and terminal objects? And why not for general C\mathbf{C}?

Posted by: Tom Leinster on December 29, 2010 3:31 AM | Permalink | Reply to this

Re: big and little toposes

Take a small category CC with both an initial and a terminal object. Then according to the definition in your comment, PSh(C)=Set C opPSh(C) = Set^{C^{op}} is a topos of cohesion

Yes. The special minimal case of the interval category C={*}C = \{\emptyset \to *\} was already discussed at cohesive site in the section Families of sets.

For XPSh(C)X \in PSh(C) we may interpret the value

X(*)=(Π 0(X)Γ(X)) X(\emptyset \to *) = (\Pi_0(X) \leftarrow \Gamma(X))

as a Π 0(X)\Pi_0(X)-indexed family of sets. The functor Γ\Gamma produces the disjoint union of all sets in the family, the functor Π 0\Pi_0 produces the indexing set. So this is the simplest notion of cohesive lumps of points : we take a set and partition it into subsets.

For general CC with initial and final object this interpretation remains true, only that now in addition for every object cCc \in C there is a notion of “hierarchy of lumps”

X(c*)=(Π 0(X)X(c)Γ(X)), X(\emptyset \to c \to *) = (\Pi_0(X) \leftarrow X(c) \leftarrow \Gamma(X)) \,,

where first the collection Γ(X)\Gamma(X) of all points is partitioned into cohesive pieces X(c)X(c) which are then further partitioned into cohesive pieces Π 0(X)\Pi_0(X).

I have added this more general discussion to the entry now.

Notice, however, that while these examples have a quadruple of adjoint functors with the leftmost one preserving (in particular) finite products, they fail to satisfy two extra axioms that Lawvere introduced later in

Bill Lawvere, Axiomatic cohesion.

One of them demands that the canonical morphism Γ(X)Π 0(X)\Gamma(X) \to \Pi_0(X) is an epimorphism for all XX. This says that “every cohesive piece of points indeed has at least one point”. This fails in the above class of examples: there are cohesive pieces with no points.

topos of cohesion (= cohesive topos?).

Yes, that’s how we ended up using the terminology, deviating a bit from Lawvere’s usage. I think “cohesive topos” fits better with “connected topos”, “local topos”, “locally connected topos”. Which is good, because a cohesive topos is all three of this.

Posted by: Urs Schreiber on December 29, 2010 10:24 AM | Permalink | Reply to this

Re: big and little toposes

Great: now I can see what the roles of the initial and terminal objects are. I’ve learned some other things too, but I can also see that there are several basic aspects of the overall situation that I don’t yet grasp. I look forward to the day that I do.

Meanwhile, here’s a thought on terminology. I prefer “topos of cohesion” to “cohesive topos”. Actually, I’m not super-keen on “topos of cohesion”; to my ears it sounds a bit clumsy. But I find “cohesive topos” actively misleading. Surely it’s the objects of the topos that are cohesive, not the topos?

(Well, maybe you can argue that a topos of cohesion is itself cohesive too. But as I understand it, that’s not the principal idea.)

Similarly, I find “smooth topos”—used to mean a topos whose objects are some kind of smooth spaces—misleading. It’s not the topos that’s smooth, right? It seems important to me that the terminology doesn’t create confusion between the attributes of a category and the attributes of the objects it contains. We don’t use “finite category” to mean a category whose objects are finite (in any given sense), or “connected category” to mean a category whose objects are connected.

Before this exchange, I knew a small amount about the distinction between big and little toposes, and toposes of generalized spaces versus toposes as generalized spaces, and I knew that Bill’s recent work on axiomatic cohesion was about this kind of thing. And if you’d asked me what the phrase “cohesive topos” meant, I would have said to myself: well, Bill uses “cohesive” to mean something like “space-like” or (in a broad sense) “geometric”, so the cohesive toposes must be the toposes that deserve to be viewed as generalized spaces. In other words, I would have got exactly the wrong answer!

Posted by: Tom Leinster on December 30, 2010 7:11 AM | Permalink | Reply to this

Re: big and little toposes

Urs wrote:

The special minimal case of the interval category[…]

That’s interesting. Is this in some sense the terminal example of a cohesive topos? Is this true when we require that pieces have points, or drop that assumption?

Posted by: David Roberts on December 30, 2010 7:47 AM | Permalink | Reply to this

Re: big and little toposes

I don’t know the answer, but another example worth considering is Set ESet^E where EE is the two-element monoid (viewed as a one-object category!) consisting of the identity and an idempotent. Thus, an object of Set ESet^E is a set equipped with an idempotent endomorphism.

A set SS equipped with an idempotent endomorphism can equivalently be described as a diagram RS R \stackrel{\leftarrow}{\rightarrow} S in SetSet exhibiting RR as a ‘split subset’ of SS. (I mean, RSR \to S followed by SRS \to R is the identity on RR; I’m too lazy to typeset a nice diagram.) Such a diagram can be expressed as a functor E¯Set\bar{E} \to Set, where E¯\bar{E} is a certain two-object category that I hope is obvious. So, Set ESet E¯Set^E \cong Set^{\bar{E}}. At a formal level, that’s because E¯\bar{E} is the Cauchy-completion of EE.

Anyway, Set ESet^E is a topos of cohesion. It even satisfies Lawvere’s extra two axioms (cohesive pieces have points, and pieces of powers are powers of pieces). Actually, the situation is kind of interesting. There are only two functors in play here: the diagonal functor Δ:SetSet E \Delta: Set \to Set^E (equipping a set with the identity endomorphism), and the functor :Set ESet \nabla: Set^E \to Set that splits the idempotent. (Or, if you’re thinking in terms of Set E¯Set^{\bar{E}} rather than Set ESet^E, it sends a diagram RSR \stackrel{\leftarrow}{\rightarrow} S as above to RR.) And the point is that

\nabla is both left and right adjoint to Δ\Delta.

That’s because when you view an idempotent as a functor ESetE \to Set, the splitting of the idempotent is both the limit and the colimit of this functor. So we get an infinitely long chain of adjunctions ΔΔ. \cdots \dashv \nabla \dashv \Delta \dashv \nabla \dashv \Delta \dashv \cdots.

I’m saying nothing more here than that this seems like an interesting example to consider.

Posted by: Tom Leinster on December 30, 2010 8:45 AM | Permalink | Reply to this

Re: big and little toposes

this seems like an interesting example to consider.

The cohesive topos that you describe may be thought of as that of skeletal reflexive directed graphs: reflexive directed graphs all whose edges have the same source and target.

To see this is is helpful to first make the cohesive topos of all reflexive directed graphs explict. I have written that out now at cohesive topos – Reflexive directed graphs (following Lawvere and you in idempotent-splitting style). This is the “1-truncation” of the cohesive topos of all simplicial sets, described further below.

We have in

(Π 0DiscΓCoDisc):RDGraphsSet (\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) : RDGraphs \to Set

that

  • ΓX\Gamma X is the set of vertices of a graph

  • Π 0X\Pi_0 X is the set of connected components of a graph (the coequalizer of the source and target maps)

  • DiscSDisc S is the graph with vertices SS and only “identity morphisms”;

  • CoDiscSCoDisc S is the “indiscrete” graph on SS: vertices SS and precisely one edge per ordered pair of vertices.

If we restrict to graphs all whose edges are loops, then evidently ΓX=Π 0X\Gamma X = \Pi_0 X. This is the case you considered (and then also the codiscrete graph of this form equals the discrete graph on the same set of vertices).

Lawvere does consider this example, albeit in maximally terse style. It is his category LL on page 6 of Axiomatic cohesion .

The cohesive topos of reflexive directed graphs was his central example already in Categories of spaces . There his point was that if one drops the reflexivity condition, then cohesiveness fails: the topose DGraphDGraph of all directed graphs is not cohesive (because Π 0\Pi_0 fails to respect products).

That old article argues that DGraphDGraph qualifies as a “generalized space” (after prop 2) while “RDGraph” qualifies as a category of generalized spaces.

Posted by: Urs Schreiber on December 30, 2010 10:33 AM | Permalink | Reply to this

Re: big and little toposes

Meanwhile, here’s a thought on terminology.

I fully agree with your arguments, I had had the same exchange with myself several times ;-)

Finding an overall consistent terminology is always difficult. Therefore here is a counter argument to your arguments and in favor of “cohesive topos” after all:

if we do regard the cohesive topos as a generalized space itself, then it is itself cohesive in a way: it behave like the archetypical cohesive lump of points for its given notion of cohesiveness.

One formal way to see that properties shared by all the objects in a topos \mathcal{E} is also shared by the topos itself is this:

to every object XX \in \mathcal{E} is associated the over-topos /X\mathcal{E}/X. This sits by an éetale geometric morphism over :(X X ):/X\mathcal{E}: (X^* \dashv X_*) : \mathcal{E}/X \to \mathcal{E}.

This generalizes the way that every object in a topos of sheaves on a topological space XX is an étale space over XX.

And this identification of objects in the topos with toposes over the topos is faithful, in that we have an equivalence of categories

(Topos/) et (Topos/\mathcal{E})_{et} \simeq \mathcal{E}

of the full subcategory of the over-category of all toposes over \mathcal{E} on the étale geometric morphisms.

So this gives a precise sense to the idea that every object XX \in \mathcal{E} is a generalized space locally equivalent to \mathcal{E} (generalizing the notion of local homeomorphism of topological spaces).

So if objects XX \in \mathcal{E} are characterized by the fact that they are locally modeled on cohesive lumps of points of sorts, then this means that \mathcal{E} itself is that cohesive lump of point.

So a topos of cohesision is itself indeed cohesive, in a sense.

Posted by: Urs Schreiber on December 30, 2010 11:35 AM | Permalink | Reply to this

Re: big and little toposes

Therefore here is a counter argument to your arguments and in favor of “cohesive topos” after all:

This terminology issue is already visible in the term “big/ little topos”: regarded as a generalized space, a big topos is very little and a little topos is very big!

Notably a big topos that is \infty-cohesive necessarily has the shape of a point and his hence maximally little as a generalized space.

This are all different aspects of the general fact that every topos is a category of generalized spaces that are locally modeled on the topos regarded itself as a generalized space. These two aspects are not independent and so terminology is always bound to oscillate between the two perspectives. I think.

Posted by: Urs Schreiber on December 30, 2010 12:22 PM | Permalink | Reply to this

Re: big and little toposes

On the other hand, it’s not true for arbitrary small categories CC that PSh(C)PSh(C) is a topos of cohesion.

Yes, a counter-example emphasized by Lawvere are the presheaf toposes GSetPSh(BG)G Set \simeq PSh(\mathbf{B}G) for GG a finite non-trivial group.

Maybe someone knows exactly which CC it is true for. (I couldn’t see it on the nnLab page.)

I don’t know. What’s the most general condition that ensures that taking the limit over a presheaf has a right adjoint?

What we have on the nnLab is a notion of cohesive site CC that encapsulates some sufficient conditions for Sh(C)Sh(C) to be cohesive. This is taylored towards examples of cohesive topose that one wants naturally in practice, such as smooth toposes modelling (synthetic) differential geometry.

(A question that I want to get the answer to is: which derived geometry gives a cohesive (,1)(\infty,1)-site?)

My questions: why does Psh(C)Psh(C) deserve to be thought of as a category of generalized spaces for any small category CC with initial and terminal objects? And why not for general CC?

This is asking: How can we tell that the axioms for a cohesive topos are good ? How can we see that indeed they do formalize the notion “category of generalized geometric spaces” in a suitable sense.

This is a good question, in particular since there is room to fine-tune the axioms. Lawvere himself has slightly modified the axioms over the years.

Questions like these find their answers by checking against loads of examples: in all the cases where we have categories that we certainly want to consider as categories of geometric spaces: do the axioms admit them? And are they sufficient to abstractly deduce all the properties of and constructions on geometric spaces that we are interested in?

It is questions like this that made me become interested in cohesive toposes: I was looking for contexts of geometry internal to which we have good general abstract notions of Lie theory and differential cohomology .

From looking at examples, I have come to the following conclusion: we should keep the core axiom (quadruple of adjoints over the base topos, with the leftmost preserving products) and discard the extra axioms (cohesive pieces have points and pieces of powers are powers of pieces). Instead, we should strengthen the core axiom by demanding that quadruple of functors not on the topos over a given site, but on the corresponding (,1)(\infty,1)-topos.

If we do this, the leftmost adjoint Π\Pi enhances from a functor that assigns connected components π 0\pi_0 to a functor that assigns the full fundamental \infty-groupoid of a generalized geometric space. Together with the fact that this has threefold right adjoints, this alone implies a pretty large amount of general structure on a cohesive (,1)(\infty,1)-topos, that seems to fully justify to thinking of this definition indeed as correctly encapsulating the notion of a category of generalized geometric spaces. A fairly long list of such structres is given at Structures in a cohesive (,1)(\infty,1)-topos.

I think this is a good proof of concept that the axiomatics of cohesive toposes is good in that it well captures the notion of categories of generalized geometric spaces.

Posted by: Urs Schreiber on December 29, 2010 12:52 PM | Permalink | Reply to this

Re: big and little toposes

Thanks for the replies. Just to nibble at this part for now:

What’s the most general condition that ensures that taking the limit over a presheaf has a right adjoint?

My guess: for a small category CC, the functor lim C op:Psh(C)Setlim_{\leftarrow C^{op}}: Psh(C) \to Set has a right adjoint iff the Cauchy-completion of CC has a terminal object.

My thinking: Because Psh(C)Psh(C) and SetSet are presheaf toposes, the existence of the adjoint is equivalent to the preservation of colimits. So, we’re looking for the small categories CC such that in SetSet, limits over C opC^{op} commute with small colimits.

(These “op”s are a bit of a pain—maybe it would be better to discuss Set DSet^D where D=C opD = C^{op}. But I’ll go with it for now.)

Essentially the only two examples I can think of are categories CC with a terminal object (because then lim C op\displaystyle\lim_{\leftarrow C^{op}} is evaluation at 11) and the one-object category consisting of a single idempotent (because then lim C op\displaystyle\lim_{\leftarrow C^{op}} is the splitting of idempotents, which is an absolute limit).

To prove the guess, I suppose one needs to use the fact that a Set-valued functor on a Cauchy-complete category is representable iff it is absolutely flat. Or maybe the closely related fact that a small Cauchy-complete category has a terminal object iff it admits a cocone on every small diagram. Or maybe something else in the same ballpark.

At the other end of the string of adjunctions, one needs Π 0=lim C op\Pi_0 = \displaystyle\lim_{\to C^{op}} to preserve finite products. This says that CC is cosifted. So all in all, I’m guessing that Psh(C)Psh(C) is a topos of cohesion iff CC is cosifted and its Cauchy-completion has a terminal object.

If that’s right, it would be interesting to know whether there’s a neater way of expressing the condition “cosifted and the Cauchy-completion has a terminal object”.

Posted by: Tom Leinster on December 29, 2010 7:23 PM | Permalink | Reply to this

Re: big and little toposes

Thanks for input, Tom!

At the other end of the string of adjunctions, one needs Π 0=lim C op\Pi_0 = {\lim_\to}_{C^{op}} op to preserve finite products. This says that C is cosifted.

Yes. At cohesive site I had originally introduced the axiom “has finite products” which implies cosifted (because I was too dumb to do the analogous proof at (,1)(\infty,1)-cohesive site without that assumption, and that proof came first, before we looked into the 1-categorical situation), but as Mike had noted there a while ago in the proof, what one needs is just cosifted (however we are still assuming the terminal object there for the right adjoint.). I have cleaned that up in the entry now.

The axioms as stated there also imply the “cohesive pieces have points”-axiom. Maybe I want to take back my dismissive comment on this axiom further above, because I now remember that a while back I was about to claim (and then maybe forgot about it) that for the cohesive (,1)(\infty,1)-toposes this axiom says that every principal \infty-bundle (defined as a principal torsor) is locally trivial, or something like this. I need to think about this again.

So much for the moment, I try to reply to the rest of what you said a little later.

Posted by: Urs Schreiber on December 29, 2010 8:36 PM | Permalink | Reply to this

Re: big and little toposes

I don’t quite agree with the title of Lawvere’s article; I think that every Grothendieck topos can be regarded as a generalized space. In particular, if a topos classifies some geometric theory, then it can be regarded as the space of all models of that theory. It’s just that some toposes (those which classify a suitable type of theory, such as the theory of local rings or local C C^\infty-rings) can fruitfully also be regarded as categories of generalized spaces.

Posted by: Mike Shulman on December 30, 2010 12:46 AM | Permalink | Reply to this

Re: big and little toposes

I don’t quite agree with the title of Lawvere’s article; I think that every Grothendieck topos can be regarded as a generalized space.

Yes, I tried to say that above, too. On the other hand and for whatever it’s worth, in that old articles Lawvere takes “generalized space” to mean “localic topos” and argues that a cohesive topos (using a slight variant of the axioms) cannot be be localic.

It’s just that some toposes (those which classify a suitable type of theory, such as the theory of local rings or local C C^\infty-rings) can fruitfully also be regarded as categories of generalized spaces.

Sure. By the way, there must be a theory-theoretic way to speak about cohesiveness. What is it?

The classifying topos for local C C^\infty-rings is cohesive. We have discussed at length the geometric interpretation of the axioms of cohesiveness. Can one give a theory-theoretic interpretation?

And can we maybe use this to say something about cohesiveness-or-not of the classifying topos for local rings?

In the foreword to the reprint of his article, Lawvere mentiones something about cohesiveness of the gros Zariski topos. But I am not sure what exactly he means to claim.

Posted by: Urs Schreiber on December 30, 2010 12:59 AM | Permalink | Reply to this

Re: big and little toposes

there must be a theory-theoretic way to speak about cohesiveness. What is it?

I don’t know! I was wondering the same thing. It’s odd, because I think the syntactic category of a theory is very rarely going to be a cohesive site; but maybe one could find a different way of presenting the classifying toposes of some general class of theories that would show that they are cohesive.

Posted by: Mike Shulman on December 30, 2010 5:27 AM | Permalink | Reply to this

Re: big and little toposes

i haven’t followed this discussion closely enough yet to understand the concept of “cohesive topos” very well or to know how much i sympathize with the attempt to develop it (or to know whether the question that i’m about to ask has already been answered), but i have a naive question about it. if cohesiveness is supposed to be somewhat antagonistic to localicness, then might this antagonism actually amount to some sort of factorization system for geometric morphisms in which the cohesive morphisms come before the localic ones?

(i’ve thought a bit about various sorts of “moore-postnikov”-like factorization systems for toposes and higher toposes, and i’m vaguely aware that other people have worked on such ideas too but i haven’t read much of their work yet.)

Posted by: james dolan on January 3, 2011 6:49 PM | Permalink | Reply to this

Re: big and little toposes

some sort of factorization system for geometric morphisms in which the cohesive morphisms come before the localic ones?

My natural interpretation of “before” would suggest that you mean a factorization AcBlCA \xrightarrow{c} B \xrightarrow{l} C where cc is cohesive and ll is localic. But there does exist a factorization system (hyperconnected, localic), and hyperconnected is not the same as cohesive (though both are a strengthening of “connected”), so it seems that that couldn’t exist. Is that what you meant?

Posted by: Mike Shulman on January 3, 2011 8:20 PM | Permalink | Reply to this

Re: big and little toposes

there are a lot of factorization systems of vaguely moore-postnikov flavor for geometric morphisms between toposes (even before getting to the case of higher toposes); i’m (roughly) asking whether there might be one that incorporates a concept of “cohesive geometric morphism” as a left half, or possibly as a left third, possibly with “localic” or some allied concept as a right half or right third. or something like that; it’s a pretty vague and open-ended question.

Posted by: james dolan on January 3, 2011 9:57 PM | Permalink | Reply to this

Re: big and little toposes

if cohesiveness is supposed to be somewhat antagonistic to localicness, then might this antagonism actually amount to some sort of factorization system for geometric morphisms in which the cohesive morphisms come before the localic ones?

Maybe to some extent: a factorization system is formed by localic geometric morphisms and hyperconnected geometric morphisms.

But I am not convinced that there is a proper antagonism between localic and cohesive, even though Lawvere did once refer to that. I think that was part of experimenting with these notions. (If I am wrong about this, I’d be happy to be educated.)

[edit: after posting this I see that Mike was a bit quicker with hitting submit]

(i’ve thought a bit about various sorts of “moore-postnikov”-like factorization systems for toposes and higher toposes, and i’m vaguely aware that other people have worked on such ideas too but i haven’t read much of their work yet.)

There is a tower of nn-localic (,1)(\infty,1)-toposes (𝒳 n)(\mathcal{X}_n) for every (,1)(\infty,1)-topos 𝒳\mathcal{X}, that plays the role of the Postnikov tower of 𝒳\mathcal{X}.

An (,1)(\infty,1)-topos over an (n,1)(n,1)-site is nn-localic. Also, for H\mathbf{H} an knk \leq n-localic (,1)(\infty,1)-topos and XHX \in \mathbf{H} an nn-truncated object (stack with values in nn-groupids) we have that the slice topos H/X\mathbf{H}/X is nn-localic. So nn-localic (,1)(\infty,1)-toposes behave like generalized nn-groupoids/nn-types to some extent.

So as far as characterization of geometries and big (,1)(\infty,1)-toposes is concerned, nn-localic-ness is a measure of “derived-ness”: for instance the ordinary big étale (,1)(\infty,1)-topos is 1-localic, but the derived big étale-(,1)(\infty,1)-topos is \infty-localic.

I don’t know how this relates nicely to cohesive (,1)(\infty,1)-toposes. But I’d be very interested in whatever one can say about this.

Posted by: Urs on January 3, 2011 8:21 PM | Permalink | Reply to this

Re: big and little toposes

A fairly long while ago, Tom had remarked on the term cohesive topos :

I prefer “topos of cohesion” to “cohesive topos”. Actually, I’m not super-keen on “topos of cohesion”; to my ears it sounds a bit clumsy. But I find “cohesive topos” actively misleading. Surely it’s the objects of the topos that are cohesive, not the topos?

(I had mentioned some possible replies.)

This here is to point out that, as I only discovered just now, Bill Lawvere himself did after all use the term “cohesive topos” previously, in 2008, after it had evolved from category of Being in his 1990 Como contribution to category of cohesion in his last 2007 article: 18 years later, back in Como, he speaks about

By the way, behind the above link is also a video of his first of three lectures – but only of the first handful of minutes.

Does anyone have more information about this 2008 Como lecture?

Posted by: Urs Schreiber on May 2, 2011 6:15 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Tom,

This was a nice present. Right now I have two suggestions. Right after you introduce the word “monic” give the definition ( f is monic if for any morphisms g and h, blah blah blah). As I was working this out, I realized that this might be better with arrows rather than functional notation, or else you would have to state which way composition runs.

The second suggestion is in your examples, put the definition of a presheave as a contravariant functor between a POSet and a ?

In both cases, I have *seen* the defs, but I don’t use them enough to know them off the top of my head.


In the case of a presheaf, I can live with a definition as a functor. But I am far too lazy to wikipedia the ancillary concepts while reading the article. So if the paper is for someone like me, who *should* know what a topos is but doesn’t, then it is helpful to give
a light bit more background.

Posted by: Scott Carter on December 29, 2010 5:25 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Hi Scott,

Thanks for your suggestions. To take the second one first:

The second suggestion is in your examples, put the definition of a presheave as a contravariant functor between a POSet and a ?

Sorry, I don’t understand. By “examples”, do you mean Examples 1.6 (examples of toposes)? Also, I’m having trouble parsing the rest of your sentence.

I thought the definition of presheaf on a category was given (albeit slightly implicitly) in Example 1.6(iv), and the definition of presheaf on a space was in Example 1.6(v). But if something’s unclear, I’ll fix it.

Right after you introduce the word “monic” give the definition

I find this very interesting. First: it’s no trouble to put the definition of monic in—but that’s not the interesting thing. The interesting thing, to me, is that it never crossed my mind that someone might (1) be comfortable with the Yoneda lemma, pullbacks, etc, and (2) be interested in a categorical introduction to topos theory, but (3) not know the definition of monic.

In case it needs saying—and this being the internet, perhaps it does—I don’t regard the definition of monic as ‘so basic that everyone should know it’ or anything snooty like that. Indeed, I wouldn’t describe any mathematical definition that way. We know what we know, and we’re interested in what we’re interested in, often for random reasons. It’s just the mixture of (1), (2) and (3) that surprises me.

But maybe I shouldn’t be surprised. I know that my own mathematical knowledge forms a strangely-shaped space, so I shouldn’t be surprised at anyone else’s.

Posted by: Tom Leinster on December 29, 2010 5:50 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Tom,

Right, my problem with the monic/epic thing is that I always have to reconstruct [(fg=fh) implies (g=h)] or [(gf=hf) implies (g=h)] from the 1-1/onto situation. As I was writing that I saw that the problem was complicated by the fact that the author doesn’t tell us which way he writes compositions. Maybe one should expect the reader to do so.

I *thought* that a presheaf is a contravariant functor from the category of posets to another category, and last night I did not remember the other category. Wikipedia says that it is a simply a contravariant functor or most often a contravariant functor from the category of open sets to the category of sets.

In terms of my own knowledge base, it is pretty spotty. So I guess I can fake pull-backs and products, and I am more-or-less willing to look up Yoneda’s lemma.

Posted by: Scott Carter on December 29, 2010 3:41 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

OK, I am better now. In 1.6.iv you do say that a presheaf is a contravariant functor to the category of sets, and you say so symbolically. My bad — I don’t read carefully on the screen.

What about if that example were written, “If A is any small category, then the {\it category of presheaves} — all contravariant functors from A to the category of sets [symbol] — is a topos.” It might very well be the case that too much information is now packed into the sentence, but it addresses my original misunderstanding.

Posted by: Scott Carter on December 29, 2010 5:03 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Scott said “I always have to reconstruct [(fg=fh) implies (g=h)] or [(gf=hf) implies (g=h)] from the 1-1/onto situation”.

I find this interesting as personally I can never remember what 1-1 and onto mean. To me, 1-1 sounds like “bijective” and I’m pretty sure that if I said “a one-to-one correspondence…” in normal life, a non-mathematician would think I meant bijection (without knowing the word).

For “onto” I have to remember that sur means on, and therefore onto means surjective. Also I have an allergic reaction to the use of “onto” as an adjective, so I have to wait until I’ve finished throwing up before I can get on with the above thought process.

Tom, perhaps if you include that kind of thing you could put it in a footnote, so that it doesn’t interrupt the flow? This reminds me of two things:

1. I’ve been playing with an e-reader (which I won’t name) whose books have a nice dictionary feature - you can hover over any word and a definition will pop up. This seems particularly good for reading books in foreign languages.

2. I wish recipe books would put basic explanations in footnotes. I know how to melt chocolate, for example, so don’t want to have to read through the instructions every time I read a recipe.

3. I am often criticised for explaining too much when I write, and dream of electronic versions of papers where one could click on something to get a fuller explanation, to save experts the oh-so-arduous task of skipping that part. Maybe other people could annotate the paper as well, adding their own bits of explanation at relevant places. It could be like google maps, where users add photos and there’s a little marker telling you where there’s some user-added content…

4. I can’t count.

Posted by: Eugenia Cheng on December 29, 2010 6:39 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Much of that is actually a goal of a group of mathematicians working together with Voevodsky: something like a combination of automated proofs and smart search engines so that readers of mathematical texts can let pop up definitions, relevant pieces of other texts, proofs, examples etc.

Posted by: Thomas on December 29, 2010 8:19 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

To me, 1-1 sounds like “bijective” and I’m pretty sure that if I said “a one-to-one correspondence…” in normal life, a non-mathematician would think I meant bijection (without knowing the word).

I’ve actually seen several textbooks which define “one-to-one” to mean injective, but “one-to-one correspondence” to mean a bijection, and I think this unfortunately reflects common usage among mathematicians.

Also I have an allergic reaction to the use of “onto” as an adjective

I think you’ve put your finger on something about that terminology that always bugged me without my clearly identifying it.

Posted by: Mark Meckes on December 29, 2010 10:17 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I think there’s even an odd usage among some mathematicians whereby “bijection” sometimes means “bijection onto its image,” i.e. injective. Perhaps this comes from the misguided idea that a function is just a set of ordered pairs, without its source and target being specified as part of the data. (-:

Posted by: Mike Shulman on December 30, 2010 5:17 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

That sounds like you would like to write math on a wiki!

Posted by: Mike Shulman on December 30, 2010 12:40 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

If only there were some place to do that.

Posted by: Tom Leinster on December 30, 2010 12:42 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

No, I want annotated research papers. I don’t think research papers should be wikis.

Posted by: Eugenia Cheng on December 30, 2010 11:52 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Posted by: Eric on December 30, 2010 3:27 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Except she said she doesn’t want a wiki, she wants annotated research papers. The idea as I see it would be to have a hyperlinked paper-tree, where the reader could “zoom in” on a particular point for more elaboration if she needed to by clicking a link.

The nLab itself could stand to take greater advantage of this zooming idea. We’ve certainly discussed it from time to time.

Posted by: Todd Trimble on December 30, 2010 12:18 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Good grief.

If you want an annotated research paper, make an annotated research paper on the nLab. It has everything Eugenia wished for. This mental barrier against the word “wiki” is a travesty I do not think I will ever understand.

There is nothing stopping her, you, or anyone else from having what you seem to want except your own willingness to accept that it already exists. The entire concept of Instiki was pioneered by Jacques for the very purpose of collaborating on research papers.

Boggled…

Edit: I see Urs comments say what I tried, but more elquently.

Posted by: Eric on December 30, 2010 1:03 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I see Urs comments say what I tried, but more elquently.

And also more politely, in a manner of speaking.

In fact, part of the purpose of this discussion will hopefully be to change people’s ideas of what a wiki can be used for. As I have already said, most people think of a wiki as something collaborative. It need not be, as by now all three of us are saying.

Posted by: Todd Trimble on December 30, 2010 2:07 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Except she said she doesn’t want a wiki, she wants annotated research papers.

What is a wiki? Nothing but the idea of a hyperlinked text realized in an easily editable fashion. Do with it what you want.

I have been writing all my research papers since the inception of the nnLab into a wiki. But since I don’t count in these discussions, here is an interesting example:

Kevin Costello is writing his research papers into a wiki. His article with Owen Gwilliam on factorization algebras is here: Factorization algebras (wiki research article).

Posted by: Urs Schreiber on December 30, 2010 12:34 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Kevin Costello is writing his research papers into a wiki.

Jacques Distler is writing his research articles on a wiki and then transforms them into arXiv postings. See his post Figures.

In the comment section to that entry, you can see that Marco Gualtieri is doing this, too.

(For those category theorists who don’t know the researchers behind these three names: all three are leading figures in their fields of research, in case that information matters. )

Posted by: Urs Schreiber on December 30, 2010 12:42 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Urs wrote:

What is a wiki? Nothing but the idea of a hyperlinked text realized in an easily editable fashion.

Okay. I think most people however, hearing the word “wiki”, assumes it means something like the nLab or Wikipedia, which allows anyone to edit, so that “ownership” is shared. (Certainly that’s the case with the nLab proper, which is where Eric linked to.) I assumed this is what Eugenia didn’t want.

Naturally I agree with you that a personal wiki can be whatever you want it to be, for example research articles. The rub is getting them counted as such!

But since I don’t count in these discussions…

I don’t understand what purpose is served by writing this.

Posted by: Todd Trimble on December 30, 2010 1:23 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Naturally I agree with you that a personal wiki can be whatever you want it to be, for example research articles.

This is an occasion to remind all the regulars here that they are all (still!) invited to ask us to have their personal wiki webs created. Password-protected or not, as desired.

This reminds me that I forgot to mention one more example: André Joyal has been writing chapters of his book into the wiki.

The rub is getting them counted as such!

One writes them on the wiki, then converts them into LaTeX and publishes them as usual. The wiki version serves the purpose of being the hyperlinked and annotated version that is useful for usage, the other version is to feed it through the current refereeing system.

Jacques Distler in the entry that I linked to says he is able to do this fully automatically: after he is done with the wiki-version, he just presses the “TeX” button (at the bottom of each instiki-page), sends the output through the LaTeX compiler and submits it to the arXiv.

For me this does not quite work as seamlessly (but Jacques and Andrew assure me that it’s my fault ;-) so I keep converting the finished wiki material to LaTeX by hand. It’s a bit more tedious, but I realize that for me personally it may actually be good, because it forces me to polish the organization further.

Posted by: Urs Schreiber on December 30, 2010 1:48 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’m still waiting for the day when we can define our own macros in itex, though.

On the other hand, it is already possible to write hyperlinked papers in ordinary TeX using HyperTeX. All the text has to be in the file in a linear order, but one could experiment with putting the “zoom text” in appendices with links to it in appropriate places from the main body. Of course the links wouldn’t be clickable in the journal version, but they would on the arXiv.

Posted by: Mike Shulman on December 30, 2010 2:19 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Concerning Tom’s request for where to have his writeup be officially peer-reviewed:

the formalities of peer review are at its core most trivial: we find and make some professional raise his or her hand and say: “This looks okay to me.” There seems to be no reason why one could not have this on any wiki.

I had suggested this a while ago on the nnForum: that we establish some simple standardized marker to put on any nnLab page saying something like “This page has been actively reviewed and found to be okay by nn readers. Click here for more details.” Where the details could be the explict names of the reviewers or if anonymity is desired it could say “A reviewer thought to be expert enough by the nnLab editorial board/steering committee”. Nothing else happens in every journal.

I have that habit of thinking these are very easy things to implement in rough form with huge payoff.

We should just start with a precedent: Tom puts his writeup into a dedicated page linked to from topos theory (if he gives me the LaTeX source I volunteer to turn it into properly hyperlinked wiki code) and one topos theory expert of the nnLab steering committee (Mike comes to mind) looks for a referee, waits for a report, and if that is positive an official stamp saying so is put on the page.

Then Tom can list this on his CV as a peer-reviewed nnLab-publication. People will of course first raise their eyebrows when seeing this, but all the more they may pay attention, have a look at the page that describes the nnLab peer review process, be convinced (if we do it right) that it is at least as reliable as any other peer review and the word will begin to spread.

Before you say “Impossible!” notice that at the wiki called the Manifold Atlas they do already have a refereeing procedure. See for instance the box at the top of the entry Aspherical manifolds, which says

This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.

Posted by: Urs Schreiber on December 30, 2010 2:58 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’m quite taken with Urs’s idea. It’s radical enough that I want to think about it for a little while first. But in any case, I also want to spend a little while improving the text following people’s suggestions.

if he gives me the LaTeX source I volunteer to turn it into properly hyperlinked wiki code

Thanks very much. I’ve actually arXived it now, so the Latex is publicly available, but let’s wait until I’ve made those improvements. (I’ll then update the arXiv version.)

I had a look at that Manifold Atlas page. I’m glad someone’s trying the idea, and I’m doubly glad that someone well-known (Wolfgang Lück) is involved. It’s very tedious when people don’t take this kind of thing seriously, and it can help to have someone senior on board.

Here’s one thing that I believe could make a big difference: an ISSN (International Standard Serial Number, like ISBN but for periodicals). It could make people more willing to submit articles. Sometimes I’ve had to list ISSN numbers on my publications list, but if I had an nnLab publication, I’d put the ISSN on my publications list whether it was requested or not—for people saying to themselves “what on earth is the nnLab?”, it would be an immediate indicator of legitimacy.

I looked into who can have an ISSN and how you get one. Being electronic-only is no problem. According to page 14 of the ISSN manual,

Serials are resources for which additional information is supplied indefinitely in a succession of discrete parts. All serials are eligible for an ISSN.

It seems to me that what Urs has in mind fits this exactly. To request an ISSN, one simply fills in a short online form.

If nnLab regulars think it’s a good idea to set this up, I could volunteer to handle the process of acquiring an ISSN. I say “I could volunteer” rather than definitely “I volunteer” because although I’m willing to do it, I’m not sure it’s proper for me to do so and then immediately submit an article.

Posted by: Tom Leinster on December 30, 2010 11:55 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Only because much that is wiki is wont to change, (maybe this was already mentioned and I missed it), it seems to me that something marked as peer-reviewed should be recoverable in the actually-reviewed form; I understand that this is in-principal doable with the changelog thing in the nlab wiki, but that doesn’t mean a generic visitor will think to do so. Is part of this implicit in Urs’ expression “dedicated page”?

Does this suggest anything to Tom/Urs/??? ?

Posted by: some guy on the street on December 31, 2010 12:48 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Here are a question and some further thoughts.

Question: would a peer-reviewed nnLab page be frozen? If not, could the stamp of approval be withdrawn?

I read a bit more about the refereeing process at the Manifold Atlas, and I see that they do create a static version once a page has been approved—but there’s also an “evolving” version. See the very bottom of the page linked.

(I also see from the editorial board that there are loads of well-known senior people involved; it’s by no means only Lück.)

Further thoughts, along the same lines as why it’s good to have an ISSN: I realized that it’s really important to people that the items on their publication list look as “official” as possible. It sucks that it matters, but it does. So I think it would be great if someone who got an article through the nnLab peer review process was then able to cite it in the following style:

The theory of theory theory. Proceedings of the nLab 1 (2011), no. 1. ISSN 12-345678.

What matters is that there are a name and a numbering system. The numbering system could be like TAC’s: 2011 = Vol 1, 2012 = Vol 2, etc. I don’t see that this would add more than the tiniest of administrative burdens, but I think it could make a big difference to those submitting articles.

The point is to make it look as traditional as possible, for the sake of reassuring conservatives—even though in some senses it’s not traditional at all. It’s so that when someone stuffy is reading your publication list, because you’ve applied for a job or a grant or whatever, they don’t dismiss your work with the words “oh, that’s just some web page”.

Posted by: Tom Leinster on December 31, 2010 12:53 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’m quite taken with Urs’s idea.

I am happy to see that you like the idea. We should try that! It may well be a small step with big payoff in the long run.

We will have to sort out the details, but I am envisioning the following procedure, which looks to me simple, robust and scalable:

we create a separate “web” (sub-part) of the nnLab

  • called Proceedings of the nnLab or something;

  • which is readable without restriction;

  • which is editable only with a password that is held by the nnLab steering committee, which functions as the editorial board until possibly a separate such board is formed.

Then anyone who wants to submit any text for publication in the nnLab Proceedings

  • creates an ordinary nnLab page with the material as usual;

  • then informs the nnLab steering committee/editorial board that he or she wishes to formally submit that page to the Proceedings

  • the board tries to go through the usual pain of finding a referee who has a close look and reports on what he or she finds, we exchange referee reports and revisions as usual and in the end…

  • if the referee approves,

    • a page on the separate password-protected web is created;

    • the content of the original nnLab page (in the approved version) is copied to that protected web;

    • a link is added to the nnLab page saying: “the version dated DD/MM/YY of this entry has been reviewed and published in the nnLab Proceedings” and provides a link.

The nnLab-version of the material is henceforth still editable as usual, but the version extracted to the separate web remains fixed…until and unless possibly the author requests to submit an update which then goes through the review process as before.

In summary: we keep a separate web not publically editable for reviewed and stably published articles. Period.

By the way, I had meanwhile started a corresponding discussion over on the nnForum .

Concerning ISSNs: I haven’t looked into this at all and would be glad if I don’t have to. It sounds like a good step to take and if you are willing to volunteer to do whatever needs to be done for this, I’d think it would be great if you did!

Posted by: Urs Schreiber on December 31, 2010 1:49 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’ll respond at the nForum… oh, no, I can’t just yet—need to wait for a moderator to approve my application for an account. In the meantime, I’ll respond here.

I’m willing to do the ISSN request, once it becomes clear that there’s enough momentum behind the overall idea. A few small decisions need making first, such as the title and the physical address of the “publisher”. If it’s thought improper for me to both do this and submit my paper, I won’t submit my paper.

Posted by: Tom Leinster on December 31, 2010 2:18 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I wrote:

I have been writing all my research papers since the inception of the nLab into a wiki.

Just for the fun of it: I have also been turning other people’s research papers into wiki pages, if I felt having such pages would be useful for me, for one reason or other:

Posted by: Urs Schreiber on December 30, 2010 1:57 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

@Eugenia Cheng

> electronic versions of papers where one could click on something to get a fuller explanation

One BIG way in how what she may be envisioning differs from current Wiki practice is that EVERY technical/defined bit of text should be hoverable/clickable.

For example every instance of the word “functor” should have a link to a page such as functor - unless that word appears within a larger phrase such as “faithful functor” in which case it should link to a page like faithful functor. Referential phrases such as “this functor” should provide some sort of link to the definition (maybe intra-pagal) of the functor being discussed.

In current Wiki practice this would be way too much work and would produce a uselessly over annotated output.

Current Wiki editing involves editorial practices such as:

1) Only provide a link for the first instance of “faithful functor” on a page.

2) Only provide a link when the concept is really important at some top level consideration such as something similar being defined or that the concept is among the most important for understanding and is not subsumed by other concepts the editor considers most important.

These practices can cause problems when:

1) Something is edited or copy-and-pasted resulting in there being no “first link”.

2) The human editor doesn’t bother to link a concept that many readers would like to be linked.

3) The reader comes across some technical term that isn’t linked and may have to search 100 lines before to where the term is defined or linked.

In a “every blessed thing is linked paradigm”

1) the display software can take care of highlighting the first link to an (important) concept on a page.

2) the editorial decision which was “what important things should have links” becomes “which links should be marked as important so that they are displayed in a highlighted fashion”.

This second decision shifts the notion of importance from particular stretches of text to the actual concepts being linked. For example if “faithful+functor” is the path to the concept/page describing it, then marking that path as being important to a page or subsection sets things up so that the first appearance of the phrase “faithful functor” on a page is highlighted as an important link without the human editor having to do anything explicitly to that stretch of text.

Such a “full linkage” approach would make the raw text source obnoxiously verbose to edit - some higher level editing software would be needed to produce it and hide the details.

One interesting aspect of full linkage is that the text strings for defined technical terms would no longer have to appear in the source of a page using them, but can be derived from the links. For example the source for a phrase like “is a faithful functor.” would not have to be ‘is a <a href=”faithful+functor”>faithful functor<a>.’ but instead could be some XML like ‘is a <t p=”faithful+functor”/>.’ and then have a display rule that uses the path “faithful+functor” to look up the text string to be displayed, and maybe a path like “faithful+functor?plural” that returns “faithful functors”.

This link-to-text aspect can be a big boon when changing terminology. Lets say I come up with some concept I initially name a “Cheng functor”. Later I determine that it comes in two flavors which I distinguish as “big Cheng functor” and “little Cheng functor”. Still later someone convincingly argues that these should really be called “top Cheng functor” and “bottom Cheng functor”. Such a renaming can be effected with just the step of redefining the link to text string maps rather than having to go through every page that mentions these concepts and edit each textual reference.

I would like to see links in Wiki like things to be used in a more “semantic” fashion rather than just as an indication of a whole page that can be displayed - as the above notion of having a mapping from a link to a text string.

Another usage of links could be in support of small (~2 sentence) concise textual definition that could appear in hover text or pop-up boxes (or more) without having to load a separate page to see the small definition. If every page that defines some concept has some way of marking part of that definition as being the small part then a link to a definition could also be used to retrieve its small version.

Furthermore the link to a small definition could be used to temporarily insert it into the page being displayed at the position where the definition was requested. That way a reader wouldn’t have to hover or pop-up to re-see the definition.

A more advanced approach to small definitions would be to parametrize them so that the definition can be passed an argument list or signature. For example if a page has the text XX is a fooCategoryfooCategory with initial element X X_&#8868; then the small definition of fooCategoryfooCategory applied to the signature (X,X )(X, X_&#8868;) would produce a text using the indicated symbols. This ability might be so useful that page writers might just be able to reference small definitions and maybe mark that they should be initially displayed rather than having to rewrite them to fit their current context.

Enough of my fantasies. I feel that this is a direction in which link usage will be moving, not something to attempt to implement now.

Posted by: Rod McGuire on December 31, 2010 11:34 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Rod said:

One BIG way in how what she may be envisioning differs from current Wiki practice is that EVERY technical/defined bit of text should be hoverable/clickable.

For example every instance of the word “functor” should have a link to a page such as functor - unless that word appears within a larger phrase such as “faithful functor” in which case it should link to a page like faithful functor. Referential phrases such as “this functor” should provide some sort of link to the definition (maybe intra-pagal) of the functor being discussed.

In current Wiki practice this would be way too much work and would produce a uselessly over annotated output.

This is a good point, but I would suggest that it can be accommodated with the current Instiki implementation by merely making links appear with the same color and font as the rest of the text so the links are not as obvious. Then we could (if wished) make every technical word a link to a corresponding page and it would not appear cluttered.

It should also be possible to toggle between different views: one with links highlighted and one without.

Enough of my fantasies. I feel that this is a direction in which link usage will be moving, not something to attempt to implement now.

I like the ideas!

For something approaching a more traditional publication, the proliferation of links can be a distraction even on the nLab currently. However, I think that before we implement new technology, we should further explore what can be done with the current technology.

I think we (as in people involved - not necessarily me) should

  1. Come up with a name, e.g. Proceeding of the nLab, nJournal, nProceedings
  2. Build a new instance of Instiki with the chosen name
  3. Experiment with a paper or two to get the technical aspects worked out
  4. Form an editorial board
  5. Obtain an ISSN
  6. Etc
Posted by: Eric on January 1, 2011 4:46 AM | Permalink | Reply to this

Re: Hyperlinked math and wikis

I presume that by “current Wiki practice” you mean “current nLab practice”? Of course different wikis may have different practices. Usually when someone says “Wiki” to refer to a particular wiki, they mean to say “Wikipedia,” so I’m excited that the nLab has reached such a “nameless” status as well… unless you actually meant Wikipedia. (-:

Only provide a link when the concept is really important at some top level consideration such as something similar being defined or that the concept is among the most important for understanding and is not subsumed by other concepts the editor considers most important.

I don’t think this is true about the nLab. My experience is that we attempt to link all terms appearing in a page, generally the first time they appear and occasionally later on as well. Sometimes people are lazy, but that is the goal. Sometimes there is an exception for concepts which are part of another concept, e.g. “faithful functor” might be linked where it first appears but not “functor”, even if this is the first appearance of “functor”—but often I think people would write [[faithful functor|faithful]] [[functor]] in that situation.

However, I do agree with the larger point that more linking could usefully be done, and I love the idea of semantic wikis.

Eric wrote:

making links appear with the same color and font as the rest of the text so the links are not as obvious. Then we could (if wished) make every technical word a link to a corresponding page and it would not appear cluttered.

I wouldn’t be in favor that; I like my links to be visibly links, so that I know not to accidentally click on them.

Posted by: Mike Shulman on January 1, 2011 7:22 AM | Permalink | Reply to this

Re: Hyperlinked math and wikis

making links appear with the same color and font as the rest of the text so the links are not as obvious. Then we could (if wished) make every technical word a link to a corresponding page and it would not appear cluttered.

I wouldn’t be in favor that; I like my links to be visibly links, so that I know not to accidentally click on them.

The links should be visible, but maybe less prominently so than currently. For instance the underline is too much. If hyperlinking is used on each and every technical term then it easily happens that half of a paragraph becomes underlined, and that looks bad.

I once experimented with changing link highlighting on my personal instiki-web, but I gave up before I reached a satisfactory solution.

Posted by: Urs Schreiber on January 1, 2011 10:20 AM | Permalink | Reply to this

Re: Hyperlinked math and wikis

The links should be visible, but maybe less prominently so than currently.

It would be nice to have a system like the one that I swear I had seen used by the NY Times: you highlight a word and a little balloon appears with a clickable link that takes you to the definition of the word in another window. This way nothing is directly clickable/linked and, on the other hand, everything is. NYT had the system that linked you to a dictionary, but dictionaries are just wikis.

Posted by: Eugene Lerman on January 1, 2011 12:17 PM | Permalink | Reply to this

Re: Hyperlinked math and wikis

Since the formatting of links really is a matter of fiddling with CSS, I think it wouldn’t hurt to explore the possibility of a simple toggle between

  • A view with links visible
  • A view with links invisible

It seems Mike and I are on opposite ends of the spectrum on this, but there is no reason both of us can’t be satisfied with a toggle.

This idea of an “nJournal” has been stewing long enough that I propose one of the nCafe authors create a new post with a title something like “Naming contest for a new higher category theory journal.”

We first settle on the name, then create a “Beta” web and start experimenting with the technicalities with a sample paper or two.

Posted by: Eric on January 2, 2011 2:05 AM | Permalink | Reply to this

Re: Hyperlinked math and wikis

Nevermind. It seems that on the nForum, a consensus has been reached on “Proceedings of the nnLab”.

Posted by: Eric on January 2, 2011 2:25 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I thought that a presheaf is a contravariant functor from the category of posets to another category, and last night I did not remember the other category. Wikipedia says that it is a simply a contravariant functor or most often a contravariant functor from the category of open sets to the category of sets.

A presheaf on a category CC is a functor

F:C opSet. F : C^{op} \to Set \,.

See nnLab: presheaf.

We call such a functor a presheaf instead of just a functor from C opC^{op} to SetSet (even though it’s the same!) to indicate that we mean to eventually equip CC with a coverage or equivalently a (Grothendieck) topology and ask whether it is in fact a sheaf with respect to that topology, in that its value on any object UU of CC is determined by its restriction to the objects U iU_i of a covering family {U iU}\{U_i \to U\} (this is what the coverage/topology specifies).

One class of examples among many of categories CC equipped with such a topology (called sites ) is the category of open subsets C=Op(X)C = Op(X) and inclusions of a topological space XX, where a family of inclusions {U iU}\{U_i \hookrightarrow U\} is called covering precisely if it is so in the naive sense.

For the purposes of topos theory it is crucial to realize that there are categories CC with topologies not of this type, and that there are categories of sheaves not equivalent to categories of sheaves on Op(X)Op(X) for some topological space XX. Those sheaf toposes that are of this form (or slightly more general) are called localic toposes.

Posted by: Urs Schreiber on December 29, 2010 3:57 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

This is very nice!

One thing which I think would make it even better would be a few comments about how the three perspectives in sections 2–4 relate to each other. To use the Elephant metaphor, you’ve given us very nice outlines of an elephant’s head, tail, and legs, but you haven’t said anything about why “it’s all one animal.”

Here are a couple of nice facts along those lines, which I think could be mentioned at the level you’re writing for.

  • The classifying topos of a theory, when regarded as a generalized space, can be thought of as the space of all models of that theory. In particular, its points are precisely the models of that theory in SetSet, while the maps into it from another space XX are the “continuous families of models” parametrized by XX.

    Some familiar spaces can be constructed in this way as the classifying topos of a natural theory. For instance, the space (or locale, or localic topos) of real numbers is the classifying topos of the “theory of a real number,” a propositional theory whose models are Dedekind cuts.

  • The way in which one constructs a category of sets violating the continuum hypothesis is to write down a geometric theory whose models are “sets intermediate in cardinality between NN and 2 N2^N,” and consider the classifying topos of that theory. Since the classifying topos of any theory contains a generic model of that theory, this classifying topos contains a “generic set violating CH,” so that CH fails in its internal logic. It isn’t quite a model of ETCS, but you can then massage it a bit to get around that (although you don’t need to, if your goal is only to prove the relative consistency of not-CH).

Posted by: Mike Shulman on December 30, 2010 12:42 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks. Those are nice facts. The one about the classifying topos of a theory as the space of models is definitely going in, and the other ones, I’ll think about.

I’m very conscious of my own ignorance here. I probably don’t know much about how the parts of the elephant are connected. To use a less surreal metaphor, maybe it’s more like a city that you’ve visited several times but don’t know well, so that you’re familiar with the layout of several neighbourhoods but don’t have a good sense of how they join together.

Posted by: Tom Leinster on December 30, 2010 1:06 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’m grateful for the feedback and I’m enjoying the discussion, but maybe I can nudge someone into helping me out with the last thing I mentioned in the post: what do I do with this article?

In the past I’ve been content to just leave stuff on the arXiv, and if that happens this time round I won’t regret writing it or anything. But these days I have a much more heightened sense of the importance of being able to say “this has been peer reviewed”. If you were rude you could call me mercenary (or worse), but I’d prefer it to put it like this: now that I’ve gone to the trouble of writing the thing, it would be nice to get a formally recognized publication out of it.

So: I’m all ears.

Posted by: Tom Leinster on December 30, 2010 9:13 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

You may wish to try

L’Enseignement Mathématique.

It publishes expository articles. The caveat is that the article should be of interest to a general mathematical audience and be accessible. Of course what that means is in the eyes of the editors.

I think your article is interesting and well written. The category theory background, on the other hand, may be a bit too demanding for a typical mathematician (in other words you assume that your readers have seen some category theory). Maybe if you add a two page appendix on adjoint functors and Yoneda, you’d be good to go.

Posted by: Eugene Lerman on December 30, 2010 12:46 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks, Eugene. I had thought of L’Enseignement, and my impression is that it’s a good journal. But rumour has it that, despite its name, it’s not interested in purely expository articles. See e.g. point 2 in Pete Clark’s MathOverflow question here, and the comments on Felipe Voloch’s answer.

(Similarly, I’ve been told that Inventiones is much more interested in the solution of old problems than the invention of new mathematics. But that’s another story.)

Have you been published by L’Enseignement? Or do you know of purely expository articles at this kind of level that they’ve published?

Posted by: Tom Leinster on December 30, 2010 11:20 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Tom,

I have two papers in L’Enseignement.

One is “Gradient flow of the norm squared of a moment map”, L’Ens Math., 51 (2005), 117-127.

The other (to appear) is Orbifolds as stacks?.

I would consider both expository as you can see from the abstracts. “Orbifolds as stacks?” was first submitted to a conference proceeding and rejected for containing nothing new. So it must be expository, no?

A few years ago I was asked by an editor of L’Enseignement to quickly look through a manuscript to see if it was truly expository and of an interest to a wide audience. So I think they take their educational mission seriously.

I also like the fact that the journal is non-profit and that their paid wall is 5 years. They have over 100 years of the journal available free on line.

Posted by: Eugene Lerman on December 31, 2010 2:10 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Eugene, do you have a more detailed reference to your orbifolds article than ‘to appear, L’Enseignement Mathématique’? I need to cite it in my anafunctors paper.

Posted by: David Roberts on January 1, 2011 10:01 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Eugene, do you have a more detailed reference…

David,

Sorry, no. I will update the arxiv entry with the journal reference as soon as I have it. I expect the paper to come out this year, 2011; the galleys went back last month.

Posted by: Eugene Lerman on January 1, 2011 12:05 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks for this very nice guide, and in particular the discussion around sober spaces and locales. I might at last have found an approach to topology I can be comfortable with!

Could you say a little more about the internal language of a topos, which you mentioned in a couple of places in the article? If I’m comfortable handling the notation of set theory and I now want to switch my attention to a topos other than SET, what do I need to bear in mind? What new rules do I have to follow when manipulating and interpreting those familiar-looking formulas in this new context, and what can I no longer do as before?

Posted by: Stuart on December 30, 2010 9:43 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

If I’m comfortable handling the notation of set theory and I now want to switch my attention to a topos other than SET, what do I need to bear in mind? What new rules do I have to follow when manipulating and interpreting those familiar-looking formulas in this new context, and what can I no longer do as before?

Essentially the single logical rule that fails is that of excluded middle: the internal logic of a general topos is intuitionistic logic.

Some discussion on how to reason inside a topos (or in a general category) is at internal logic.

The examples-section describes basics of internal logic in a presheaf topos and highlights how the failure of excluded middle arises.

Posted by: Urs Schreiber on December 30, 2010 10:50 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Thanks!

Posted by: Stuart on December 30, 2010 1:53 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

A couple of other comments:

perhaps the totality of Grothendieck toposes is not as nice as the totality of all toposes.

I would be inclined to say that the 2-category of Grothendieck toposes and geometric morphisms is also nicer than the 2-category of elementary toposes and geometric morphisms. For instance, the former has limits and colimits and classifying toposes, while the latter does not always. (However, the 2-category of elementary toposes bounded (= “relatively Grothendieck”) over any base elementary topos S is almost as nice as those bounded over Set.) The 2-category of elementary toposes and logical functors is a totally different beast, probably just as nice in its own right but of course used for totally different purposes.

How much has been lost by passing from topological spaces to locales? In most people’s view, not much.

While I share this view, it might also be nice to mention briefly what has been gained. We lose the non-sober spaces, which arguably are not very important (although non-sober topologies do arise in “real” mathematics, such as the Scott topology—Peter Johnstone has a lovely 2-page paper entitled “Scott is not always sober”). But we gain all of the non-topological (or non-spatial) locales. This has some interesting consequences, which you may be aware of:

  • The intersection of two dense sublocales is always again dense. For instance, the intersection of the rationals and the irrationals is a dense (and in particular nontrivial) sublocale of the reals. Of course, it has no points. In fact, every locale has a smallest dense sublocale (which is usually wildly non-spatial).

  • There are lots of other interesting locales with no points, such as the locale of all bijections between \mathbb{N} and \mathbb{R}. The general rule is that if there’s an object which doesn’t exist, but which you can’t prove not to exist by a finite number of “observations,” then there is a pointless but nontrivial locale (or classifying topos) of such. Topoi of sheaves on this sort of locale are often the starting place for set-theoretic forcing arguments, like the independence of the continuum hypothesis.

  • The inclusion of locales into toposes, since it is a right adjoint, preserves all limits. But the functor Open\mathbf{Open} from topological spaces to locales (or equivalently, the soberification functor) does not preserve all limits. It doesn’t even preserve finite products in general: the locale product ×\mathbb{Q}\times\mathbb{Q} (where \mathbb{Q} has the subspace topology from \mathbb{R}) is non-topological, and “bigger” than the product as topological spaces. Because of this, (,+)(\mathbb{Q},+) is not a localic group: the addition map ×\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q} is defined only on the topological-space product, not the locale product. (Products of locally compact spaces are preserved, however.)

  • This is an instance of an amazing and general phenomenon: every (localic) subgroup of a localic group is closed! Thus, since \mathbb{Q} is not a closed sublocale of \mathbb{R}, it cannot be a localic subgroup. I believe that non-closed subgroups of topological groups are often regarded as pathological—for instance, in equivariant algebraic topology, the orbit category of a topological group is defined only using closed subgroups. So I find it quite striking that by moving to locales, the non-closed subgroups simply evaporate. Perhaps this is more evidence for your “attitudinal paradox.”

  • Another amazing phenomenon, which I am also reminded of by thinking about the “attitudinal paradox,” is that the category of “locales internal to the topos Sh(X)Sh(X)”, for some locale XX, is equivalent to the slice category of (external) locales over XX. In other words, if you’re interested in “parametrized spaces” (spaces over base spaces), as some algebraic topologists are, then if you replace “space” by “locale”, you can study them by simply studying ordinary locales—as long as you do it constructively, so that you can then interpret it internally in Sh(X)Sh(X) where XX is your base locale.

Probably it would take you too far afield to say even a fraction of this in the paper, but it might be nice to convey some idea of the difference between locales and sober spaces.

Posted by: Mike Shulman on December 31, 2010 7:22 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Alex Simpson has a spectacular application of locale theory to the theory of random sequences.
He constructs a (pointfree) locale of random sequences, sequences satisfying all probabilistic (=measure 1) laws. It is easy to see that the locale has no points, still the locale itself has many interesting properties.

Posted by: Bas Spitters on December 31, 2010 9:36 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

the category of “locales internal to the topos Sh(C)”, for some locale XX, is equivalent to the slice category of (external) locales over XX.

Is there an analogous statement for locales internal to a non-localic topos?

There is the “noncommutative Gelfand theorem” for a noncommutative C *C^*-algebra

AA which realizes AA as an internal commutative C *C^*-algebra in the presheaf topos PSh(CSub(A))PSh(CSub(A)) over the semilattice of commutative subalgebras of AA and finds a locale internal to the topos that is Gelfand-dual to the internal AA.

Posted by: Urs Schreiber on December 31, 2010 10:11 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I am not sure about the non-localic version of this result.
However, the topos of presheaves over the poset C(A) of commutative subalgebras is in fact localic and hence a similar theorem applies. It is in p53 of these slides. The explicit computations can be found here. We are working on a version with a more fiber-wise presentation.

Bas

Posted by: Bas Spitters on December 31, 2010 1:15 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

the topos of presheaves over the poset C(A) of commutative subalgebras is in fact localic

Ah, thanks!

It is in p53 of these slides.

Please help me, I am not sureif I am following. What is YY and XX in 52? How do they relate to AA? In 53, we should have 𝒯(A)\mathcal{T}(A) and 𝒞(A)\mathcal{C}(A) instead of T(A)T(A) and C(A)C(A), I suppose? Could you say in plain words for me how the statement in 53 says that 𝒯(A)\mathcal{T}(A) is localic? (Sorry.)

Posted by: Urs Schreiber on December 31, 2010 2:29 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Perhaps the non-localic version you want is that for any topos E, internal locales in E are equivalent to toposes equipped with a localic geometric morphism to E? That’s certainly true: just carry out the proof of the equivalence between locales and localic topoi internally to E.

Also, any topos of sheaves on a posite (such as presheaves on a poset) is localic.

Posted by: Mike Shulman on December 31, 2010 6:31 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

any topos of sheaves on a posite […] is localic.

Thanks, that’s what I needed. I have added that remark to semilattice of commutative subalgebras.

Posted by: Urs Schreiber on January 1, 2011 1:29 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Also, any topos of sheaves on a posite (such as presheaves on a poset) is localic.

I see I was being dense. This is a trivial statement. Sorry for the confusion.

Posted by: Urs Schreiber on January 1, 2011 12:21 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Mike wrote

…every (localic) subgroup of a localic group is closed! Thus, since \mathbb{Q} is not a closed sublocale of \mathbb{R}, it cannot be a localic subgroup. I believe that non-closed subgroups of topological groups are often regarded as pathological – for instance, in equivariant algebraic topology, the orbit category of a topological group is defined only using closed subgroups.

Is this linked to the definition of a Klein geometry as a pair, a Lie group and a closed Lie subgroup? So with a localic Lie group (smooth localic group?) we could do Klein geometry without worrying about closedness of subgroup?

Posted by: David Corfield on January 1, 2011 11:59 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

So with a localic Lie group (smooth localic group?) we could do Klein geometry without worrying about closedness of subgroup?

I haven’t thought about it, but it seems reasonable.

Posted by: Mike Shulman on January 1, 2011 3:29 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

So with a localic Lie group (smooth localic group?) we could do Klein geometry without worrying about closedness of subgroup?

I haven’t thought about it, but it seems reasonable.

I’d think it is probably reasonable to expect that large parts of theory where topological spaces are considered up to homeomorphism in geometry have a more natural reformulation in terms of locales.

Analogous to, but at the other end of the spectrum, how theory where topological spaces are considered up to weak homotopy equivalence is more naturally reformulated in terms of \infty-groupoids.

Given that locales are (0,1)-toposes and \infty-groupoids are (,0)(\infty,0)-categories this says that large parts of geometric math are naturally reformulated in terms of (n,1)(n,1)-topos theory.

I’d expect that it pays to take this seriously, as is done in Structured Spaces , where the notion of locally ringed topological space is entirely replaced by locally ringed nn-localic (,1)(\infty,1)-toposes.

It seems to me that all of “geometry” is naturally formulated in these terms, eventually.

Posted by: Urs Schreiber on January 1, 2011 7:54 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Is there a condition on (n,r)(n, r)-toposes corresponding to an ordinary topos being a Grothendieck one? And, if so, could they be thought of as classifying toposes for a (n,r)(n, r)-geometric theory, whatever that means?

Posted by: David Corfield on January 2, 2011 10:14 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Is there a condition on (n,r)(n,r)-toposes corresponding to an ordinary topos being a Grothendieck one?

Currently the existing definitions for n3n \geq 3 and r=1r =1 that I am aware of (namely those going back to Charles Rezk as developed by Jacob Lurie) are only for Grothendieck toposes.

On the other hand, there is a proof that these are characterized by higher analogs of Giraud’s axioms . So for whatever definition of elementary (n,1)(n,1)-topos that you come up with, we can characterize the Grothendieck (n,1)(n,1)-toposes among them as being those that satisfy Giraud’s axioms.

if so, could they be thought of as classifying toposes for a (n,r)(n,r)-geometric theory, whatever that means?

I’d think generally the answer to this must be “yes, by definition”. But more concretely, classifying (,1)(\infty,1)-toposes for (,1)(\infty,1)-geometric theories of “local (,1)(\infty,1)-algebras” are discussed and referenced here.

Posted by: Urs Schreiber on January 2, 2011 11:25 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’d think generally the answer to this must be “yes, by definition”

I would answer, rather, “hopefully, yes, once we define a geometric (n,r)-theory.” As usual, I would prefer to use “theory” in a more syntactic sense. That seems to me to be the real fruit of the usual theory of classifying 1-toposes: the correspondence between syntax and semantics. If you define a geometric theory to be “a thing which has a classifying topos” (or “a category in the doctrine of Grothendieck topoi”, to make it sound more like the language used by the people who take “theory” to mean a certain kind of category) then tautologically, of course every geometric theory has a classifying topos. But the interesting thing is to prove that if you’re given a more syntactic sort of theory, you can construct a classifying topos, and conversely from any Grothendieck topos you can write down a syntactic theory which it classifies.

Posted by: Mike Shulman on January 2, 2011 10:49 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Hi Mike. You say it’s more interesting to take a manifestly syntactic definition of theory. But is it actually more useful? For instance, consider categories that are monadic over the category of sets. These can be described syntactically using the language of universal algebra, that is, operations of various arities and relations between them. But (perhaps like Urs) in my experience so far, I’ve always preferred to leave the language of universal algebra at an informal level. So, if I were teaching a class, I would explain how to construct free objects, and I would probably use terms like ‘alphabet’, ‘words’, ‘relations’, and so on, and I might even say a bit about why all monadic categories over the category of sets arise in such a way. But I would probably wouldn’t take the trouble to make these concepts precise for the purpose of making that statement a theorem.

I guess seems a bit like making a special tool to pick up your shoes–you could make one, but it’s easier to just do it without one. But if I’m too limited in my outlook, I would love to know why!

Posted by: James on January 3, 2011 6:34 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I’m not sure what you’re saying. When you define a group in an abstract algebra class, don’t you define it by saying there is a binary operation and a unary operation and a constant which satisfy some axioms? That’s what I call an explicit syntactic presentation. The corresponding “classifying-category” definition would require you to construct more-or-less explicitly the Lawvere theory of groups, which is a category with countably infinitely many objects, and define a group to be a product-preserving functor from that category to Sets. That definition may be more convenient for theoretical reasons, but not for others. So it sounds to me like your “special tool to pick up your shoes” describes better the second definition, whereas I’m advocating for the importance of the first (and the study of the way that you can get from the first definition to the second).

Posted by: Mike Shulman on January 4, 2011 7:18 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

When you define a group in an abstract algebra class, don’t you define it by saying there is a binary operation and a unary operation and a constant which satisfy some axioms?

However, we were not discussing if it is easier to define a particular theory syntactically or by means of its classifying topos. Instead, we were discussing how we would define the theory of theories: how we would recognize that something is an (n,1)(n,1)-theory, to wit.

If I go to the entry essentially algebraic theory and compare the two definitions there, I see that the first definition is dead simple. “A category with finite limits. A model of it is a finite limit-preserving functor. Period.” Whereas the second one looks considerably more complicated. If I had to choose, I would teach the first one in class.

Posted by: Urs Schreiber on January 4, 2011 7:59 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Mike, I guess what I’m saying is that I’ve never had a use for the general definition of theory, in the syntactic sense. Of course there are particular theories, such as the theory of groups, which I have cared about. (This might be like some applied mathematicians, who have never needed the general concept of a group, even though they use invertible matrices all the time.) It’s not really a big deal, but occasionally when I’m looking at something on topos theory and I skip over the chapters with the word ‘theory’ in the title, I wonder if I’m missing anything I’d care about.

Posted by: James on January 5, 2011 10:00 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

The point I’m making is that while the walking-category definition of a “theory” may be simpler to work with when studying theories in general, if you ever want to actually apply the “theory of theories” to deduce conclusions about any particular theory, then you need to make contact with the syntactic way in which particular theories (such as the theory of groups) are usually presented.

Posted by: Mike Shulman on January 5, 2011 7:12 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Do you mean to tell us, Jim, that you don’t want one of these?

Posted by: Tom Leinster on January 4, 2011 7:39 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Ha! That looks like it’s more for putting shoes on than picking them up, but it is true, I don’t want one.

Posted by: James on January 5, 2011 10:04 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

It’s called the Shoe Grabber, but I concede that it does have a handy built-in shoe horn. Yours for only NZ$16.99.

Posted by: Tom Leinster on January 5, 2011 10:30 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

I guess seems a bit like making a special tool to pick up your shoes–you could make one, but it’s easier to just do it without one.

For some purposes it may be useful to have that tool, though. But how do we decide that the tool works? Simple: we check if does indeed pick up the shoes!

That seems to me to be the crux of the situation: I trust Mike that it would be interesting to define the meaning of “(geometric) (n,1)(n,1)-geometric theory” in a syntactic way (and it seems that people following Voevodsky and using higher identity types in type theory are heading roughly in this sort of direction, if I understand well). But I would think that if you come up with a syntactic definition of geometric (n,1)(n,1)-theory whose models are not classified by an (n,1)(n,1)-topos, then we would conclude that you need to fine-tune your definition better. Wouldn’t you agree?

The whole situation seems to me to be a special case of a common general phenomenon:

  1. we have a relatively complicated but important concept;

  2. we find that it has dually a very elegant and simple equivalent reformulation;

  3. we take the elegant simple reformulation and generalize it vastly (because that’s elegantly and easily possible here);

  4. we declare that the vast generalizaiton of the originally fairly complicated concept is the formal dual to this generalization of the elegant simple concept.

That does not mean that the more complicated-looking incarnation is not interesting. But that we use the dually elegant simple definition to cross-check what it should be.

That sort of thing happens all the time.

Posted by: Urs Schreiber on January 3, 2011 12:48 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Hmm, here we go with this discussion again. (-:

if you come up with a syntactic definition of geometric (n,1)-theory whose models are not classified by an (n,1)-topos, then we would conclude that you need to fine-tune your definition better. Wouldn’t you agree?

Probably. But I hesitate to make pronouncements like that in advance.

we declare that the vast generalizaiton of the originally fairly complicated concept is the formal dual to this generalization of the elegant simple concept…. That sort of thing happens all the time.

Indeed. But it’s also not uncommon for people to misapply that procedure and come up with the wrong generalization, which later has to be corrected by people who think more carefully about the more general situation. To take a couple of relevant examples:

  • A category is a category enriched over sets. A 2-category is a category enriched over categories. Surely an n-category is a category enriched over (n-1)-categories? But no, that gives us only strict n-categories, which aren’t very useful for n>2n\gt 2.

  • Sheafification can be performed in two steps with the plus-construction, or in one step using hypercovers. Similarly, stackification can be performed in three steps with a plus-construction or in one step using hypercovers. Surely the best way to do \infty-stackification is using hypercovers (since a plus construction doesn’t then necessarily even converge after ω\omega-many steps)? But no, that gives us only hypercomplete (,1)(\infty,1)-topoi, which aren’t as well-behaved and miss out on some important phenomena.

So I think it’s important to approach each situation on its own merits, guided by our hopes and expectations from situations that we’re generalizing, but not blindly defining things to be the generalizations that we hope to prove that they are.

Posted by: Mike Shulman on January 4, 2011 7:27 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

my attitude towards grand cryptomorphisms is that the whole point of one is that once you’re in on the secret, you don’t have to bother learning both of the subjects that it bridges; you just learn the one that you prefer and pretend to be an expert on the other too by exploiting the cryptomorphism. for extra effect you may tell people that you’re thinking about both subjects at the same time but this is often best accomplished by redefining the foreign concepts in terms of the familiar, allowing fluent enough communication with people on the other side of the cryptomorphism so that genuine cross-fertilization can happen in both directions.

the concept of “coherent geometric theory” seems to me like an example of a concept with one foot on either side of the bridge between theories-as-categories and theories-as-theories. to me a more natural or at least more obvious kind of category is one with all finite limits and all finite colimits, the limits distributing over the colimits the way they do in the category of sets. this is parallel for example to the concept of “abelian category” as a category with all finite limits and all finite colimits, the limits distributing over the colimits as they do in the category of abelian groups.

but the syntactic category of a coherent geometric theory, if i understand correctly, generally has all finite limits but only some finite colimits; in particular it has coequalizers of equivalence relations but not more general coequalizers. the colimits here can be thought of as “secondary operations”, well-defined on a domain defined in terms of the “primary operations” given by the limits. i suspect that this means that the forgetful (2,1)-functor from coherent geometric theories to categories is a composite of two monadic (2,1)-functors but is not itself monadic (much as with the forgetful functor from categories to sets, the operation of morphism-composition being a secondary operation well-defined on a domain (the “composable strings” of morphisms) defined in terms of the primary operations of source and target). if (as i optimistically presume) the idea that locally finitely presentable categories can be thought of as those given by finite-limit-sketches can be appropriately generalized to the (2,1) context, then this should be enough to show that the (2,1)-category of coherent geometric theories is locally finitely presentable.

but even if (as i’m trying to suggest above) coherent geometric theories can profitably be studied from the theories-as-categories viewpoint, i suspect offhand (not knowing very much about the history of the concept) that a lot of the motivation for the concept comes more likely from the theories-as-theories viewpoint (or at least from the interaction between the two viewpoints).

(i have the vague impression that “horn logic” provides another example of a sort of logic which invites expression in terms of syntactic categories, but of a kind which you might not have thought of, coming from a more natively category-theoretic viewpoint.)

Posted by: james dolan on January 3, 2011 9:44 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

That sounds reasonable. Except that I would be a little more inclined to say that the syntactic category of a coherent theory is a coherent category, which has even fewer colimits than that (e.g. only coequalizers of equivalence relations which occur as kernel pairs, and only coproducts of pairs of objects which can be embedded disjointly in a third object). However, pretopoi are a reflective subcategory of coherent categories, so for purposes of looking at models in topoi, any coherent category can be “completed” to a pretopos in a universal way.

Posted by: Mike Shulman on January 4, 2011 8:30 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Mike wrote (lots of very interesting things and then)

Probably it would take you too afar afield to say even a fraction of this in the paper, but it might be nice to convey some idea of the difference between locales and sober spaces.

Yes… it’s always a subjective judgement, how much to include. One can of course decide that one is always going to include that interesting thing just over the horizon, but in that case one’s text is named after a pachyderm.

I probably feel that it’s enough to have introduced locales, pointed out the adjunction between locales and toposes, pointed out the adjunction between spaces and locales, and said a little about the equivalence to which the latter adjunction restricts. That seems like enough to absorb at a single sitting. But perhaps I’ll add a sentence tempting the reader to go and read about the kind of things you mentioned in your comment.

I already have the feeling that I included slightly too much. In Latex it’s 24 pages, and I would have liked to have kept it strictly under 20. But as I wasn’t willing to either condense the writing style or cram more onto each page, that would have meant chopping some material. So I decided to leave it.

Posted by: Tom Leinster on January 4, 2011 10:21 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Mike wrote:

I believe that non-closed subgroups of topological groups are often regarded as pathological

The quotient of a topological group by a subgroup is Hausdorff if and only if the subgroup was closed. (Reference here.) Given that people often see non-Hausdorffness as pathological, it would be natural if non-closed subgroups were also seen as pathological.

Posted by: Tom Leinster on January 4, 2011 10:42 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Mike wrote:

I believe that non-closed subgroups of topological groups are often regarded as pathological

Non-closed Lie subgroups are sometimes referred as “virtual subgroups” and are treated gingerly. You need them if you want every Lie subalgebra of a Lie group to correspond/define a Lie subgroup. On the other hand you have to regard them as immersed rather than embedded, otherwise they are not manifolds.

I suspect that what bothers people is that in the category of manifolds the correct notion of a subobject is that of an injectively immersed submanifold, not an embedded submanifold.

Posted by: Eugene Lerman on January 5, 2011 12:38 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Still trying to get a grip on what we can do with just locales rather than topological spaces…

For example, since topological manifolds are defined in terms of (sober) topological spaces and local homeomorphisms, does that mean we can adapt the definition to refer instead to locales – “a locale equipped with maps to Open( n\mathbb{R}^n) such that …”? If someone hands us a topological manifold and we locale-ize it in this way, can the original manifold be recovered or has some information been lost? Is there a name for the kind of generalized smooth space that completes the analogy

topological space : locale :: topological manifold :???

And can we suitably add extra structure to get locale-ized versions of the more interesting kinds of manifolds, such as Lorentzian or Kähler manifolds?

Posted by: Stuart on January 1, 2011 8:32 AM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Is there a name for the kind of generalized smooth space that completes the analogy topological space : locale :: topological manifold : ???

Yes… “topological manifold”. (-: Since open subsets of n\mathbb{R}^n have enough points, if you glue them together along opens you will still get something having enough points. More generally, any locale that admits a surjection from a topological space is itself topological.

(Interestingly, that means that “non-Hausdorff manifolds”, like the line with a doubled point, give some nice examples of sober but non-Hausdorff spaces.)

Note, though, that if you work constructively, then even the locale \mathbb{R} need not have enough points! The topological space of real numbers can have some poor properties (e.g. the Heine-Borel theorem can fail) but this is not true if you work with the locale instead. So in that case it might be the “localic manifolds” which are what you want to work with instead. I don’t know whether anyone has tried putting Lorentzian or Kahler structures on them.

Posted by: Mike Shulman on January 1, 2011 3:34 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Tom Leinster is planning to submit a revised version of his article An informal introduction to topos theory for peer review to the Annals of the nnLab.

This requires that the document be converted to an nnLab page first. A skeleton of the LaTeX-to-instiki conversion is at

This can be continued, and done in small parts at a time, by anyone who is interested, for one reason or another. One reason might be that you are a student lurking here, would benefit personally from going through the text carefully, and would enjoy the eternal fame of being an nnLab co-author.

(Even more eternal fame for anyone who starts writing a script that would help automate this process.)

For more details see the HowTo.

Posted by: Urs Schreiber on January 4, 2011 2:44 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Submitted now. It’s this Latex version that I’ve submitted for peer review. Andrew Stacey has written a script for converting Latex to Instiki, and he’s going to run that on my revised version (thanks, Andrew!), so please don’t spend time cleaning up the version that’s currently on the Lab: it’ll be replaced.

Urs mentioned coauthorship. I’m not sure what he had in mind, but in my mind this isn’t a possibility except in the case that a referee has a large amount of input. The paper submitted for peer review is single author, and while of course I’m extremely grateful to those who are helping with the conversion process, I don’t think this is a reason to be named as a coauthor.

(For comparison: every now and then, Reprints in Theory and Applications of Categories runs a project whereby a team of people type up some out-of-print classic category theory text. I’ve been involved in a couple of these, and it’s a fair amount of work. But the typists aren’t named as authors.)

I’m aware that this doesn’t sound particularly generous, but I think it’s fair; and the system will collapse if it’s not perceived to be fair.

If I understand the process correctly, we’re going to convert the Latex document into wiki format while the peer review is going on. The wiki page should be an exact replica of the Latex document, modulo formatting. (In some ways it will be better, because it’ll be hyperlinked.) If the paper passes peer review then—after the necessary revisions according to the referee report(s)—that replica page will be frozen, stamped “approved”, and write-protected. A duplicate page will also be created, which will be like any other nnLab page: anyone can edit it, and it takes on a life of its own, beyond my control. But it’s important to me to have complete control of the peer-reviewed version of the text, as in an ordinary journal submission.

Posted by: Tom Leinster on January 7, 2011 7:23 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Urs mentioned coauthorship. I’m not sure what he had in mind

By nnLab coauthor I meant what is maybe better called contributor . I was just trying to sell the idea that people might help with converting the source, not that they become coauthors of the peer-reviewed version of your article that will be moved to the nnJournal.

Posted by: Urs Schreiber on January 7, 2011 7:59 PM | Permalink | Reply to this

Re: An Informal Introduction to Topos Theory

Ah, I see. Sorry; I misunderstood.

Posted by: Tom Leinster on January 7, 2011 8:03 PM | Permalink | Reply to this

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