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November 22, 2011

Discreteness, Concreteness, Fibrations, and Scones

Posted by Mike Shulman

Today I realized that two old friends of mine are closely related: categories of spaces with discrete and codiscrete objects, and the monadicity of fibrations and opfibrations. The glue between them is called the scone.

Let’s start in a setting that’s hopefully comfortable: a forgetful functor U:CSU\colon C\to S. We want to think of SS as a category of “unstructured” set-like objects (or perhaps algebraic objects), and CC as a category of “spaces” over it. That is, an object of CC should be thought of as an object of SS equipped with some sort of “space-structure”, “topology”, or “cohesion”.

For instance, if SS is sets, then CC could be topological spaces, convergence spaces, subsequential spaces, locales, the category of sheaves on a site like CartSp or Diff or Top, or Johnstone’s topological topos. We don’t assume that UU is faithful in general, but we may as well assume that it is an isofibration.

We say that CC has discrete objects if UU has a fully faithful left adjoint, and codiscrete objects if UU has a fully faithful right adjoint. By abstract nonsense, if UU has both adjoints, then one is fully faithful if and only if the other is so. On the other hand, we might also wonder whether UU is a Grothendieck fibration or opfibration. In fact, these are closely related.

Theorem 1: Suppose CC has a terminal object preserved by UU. If UU is a fibration, then CC has codiscrete objects.

Proof: Define G:SCG\colon S\to C by G(a)=G(a)= the result of pulling back the terminal object 1C1\in C along the unique map a1a\to 1 in SS. It is easy to verify that UGU\dashv G and that GG is fully faithful. \Box

Theorem 2: Suppose CC has pullbacks preserved by UU. If CC has codiscrete objects, then UU is a fibration.

Proof: Assuming UGU\dashv G with GG fully faithful, and given xCx\in C with U(x)=bU(x)=b and a morphism f:abf\colon a\to b in SS, consider the pullback f *x Ga x Gb\array{f^\ast x & \overset{}{\to} & G a \\ \downarrow && \downarrow\\ x & \underset{}{\to} & G b} Since this pullback is preserved by UU, we have U(f *x)=aU(f^\ast x) = a. (Or at least U(f *x)aU(f^\ast x) \cong a, and since UU is an isofibration we can choose f *xf^\ast x to make this an equality.) The universal property of a cartesian arrow is again easy to verify. \Box

Thus, if CC has finite limits preserved by UU, then it has codiscrete objects if and only if it is a fibration. Dually, of course, if CC has finite colimits preserved by UU, then it has discrete objects if and only if it is an opfibration. (More generally, if CC is complete or cocomplete and UU is continuous or cocontinuous, then we can construct “final lifts of small UU-structured sinks”.)

Now suppose that CC lacks one or both of discrete and codiscrete objects; how can we modify it so that it will have them? One idea is to construct a new category whose objects are explicitly “objects of SS equipped with CC-structure”. Specifically, we consider the category of triples (aS,xC,aU(x))(a\in S, x\in C, a\to U(x)) (that is, the comma category of Id SId_S over UU).

In this context, we call this category the scone (short for Sierpinski cone) of CC over SS, or scn S(C)scn_S(C). It comes equipped with obvious functors U:scn S(C)SU'\colon scn_S(C)\to S and i *:scn S(C)Ci^\ast \colon scn_S(C)\to C. We also have a functor i *:Cscn S(C)i_\ast \colon C\to scn_S(C) defined by i *(x)=(U(x),x,id U(x))i_\ast (x) = (U(x),x,id_{U(x)}), which is right adjoint to i *i^\ast .

Intuitively, aa is a set, xx is a space, and aU(x)a\to U(x) says that each element of aa corresponds to a point of xx. If this morphism is not injective, then our new object has “multiple points that can’t be told apart by the topology”, while if it is not surjective, then our new object “has room for more points in the topology than are actually present”. This suggests the following.

Theorem 3: If CC has a terminal object preserved by UU, then U:scn S(C)SU'\colon scn_S(C)\to S has a fully faithful right adjoint, which takes aSa\in S to (a,1,!:aU(1))(a,1,!\colon a\to U(1)). Thus scn S(C)scn_S(C) has codiscrete objects.

Proof: Easy. \Box

Moreover, Theorem 3 is a corollary of Theorem 1, because UU' is always a fibration: for f:abf\colon a\to b we have f *(b,x,bU(x))=(a,x,afbU(x))f^\ast (b, x, b\to U(x)) = (a,x, a\xrightarrow{f} b \to U(x)). In fact, UU' is the free fibration generated by UU: the category of fibrations over SS is 2-monadic over Cat/SCat/S, and UUU\mapsto U' is the 2-monad with unit i *i_\ast .

This 2-monad is colax-idempotent, so that UU is itself a fibration if and only if i *i_\ast has a right adjoint i !i^! that commutes with UU and UU'. Therefore, from theorems 1 and 2, we conclude:

  • If CC has a terminal object preserved by UU and i *i_\ast has a right adjoint over SS, then CC has codiscrete objects. This is easy to see directly by composition of adjoints, since U=Ui *U = U' \circ i_\ast and UU' always has a right adjoint.

  • If CC has pullbacks preserved by UU and codiscrete objects, then i *i_\ast has a right adjoint over SS. The adjoint is defined by the pullback i !x G(a) x GU(x)\array{i^! x & \overset{}{\to} & G(a)\\ \downarrow && \downarrow\\ x & \underset{}{\to} & G U(x) }

Thus, if we restrict to the category of lex categories and lex functors over SS, then we can also regard UU' as the free category-with-codiscrete-objects generated by UU.

Dually, of course, we can consider the “co-scone” which is the free opfibration and the free category-with-discrete-objects. However, we also have the following nice fact.

Theorem 4: If UU has a left adjoint, then so does UU', which is fully faithful if the left adjoint of UU is so. Thus if CC has discrete objects, so does scn S(C)scn_S(C).

Proof: Let FUF\dashv U; we define F:Sscn S(C)F'\colon S\to scn_S(C) by F(a)=(a,F(a),aUF(a))F'(a) = (a, F(a), a\to U F(a)). The universal property is easy to verify. \Box

This means that there must be a distributive law relating the scone and the co-scone, enabling us to talk about joint algebras for the two monads. These joint algebras are, of course, functors into SS which are both fibrations and opfibrations, or equivalently (in the lex and colex case) those having both discrete and codiscrete objects.

Let’s bring it all together by recalling two important examples. Firstly, suppose that CC and SS are toposes and UU is the direct image part of a geometric morphism (thus it has a left-exact left adjoint). Then by Theorem 4, UU' also has a left adjoint, which inherits left-exactness; thus UU' is also the direct image part of a geometric morphism. Finally, i *i_\ast always has a left-exact left adjoint i *i^\ast , so the morphism i:Cscn S(C)i\colon C\to scn_S(C) lives in Topos/STopos/S.

In this case, having codiscrete objects (which then implies also having discrete ones) is called being a local SS-topos. The fact that scn S(C)scn_S(C) is the free local SS-topos on CC appears in C3.6.5 of Sketches of an Elephant. Theorem 2 implies that for any local SS-topos, the “global sections” morphism U:CSU\colon C\to S is a fibration and opfibration, a useful thing to know. Note that in this case UU is not generally faithful.

Secondly, let S=SetS=Set and C=LocC=Loc be the category of locales, with UU the “set of points” functor (also not faithful). Then scn S(C)scn_S(C) is the category of topological systems defined in Steve Vickers’ book Topology via Logic. These are “midway” between topological spaces and locales, having both a frame of opens and a set of points, neither of which is necessarily determined by the other.

There is also a way to recover the usual category of topological spaces. If U:CSU\colon C \to S has codiscrete objects, we say that xCx\in C is concrete if xGU(x)x\to G U(x) is a monomorphism. This is equivalent to saying that UU is faithful on morphisms with codomain xx. Dually, if CC has discrete objects, we say xx is co-concrete (“ncrete”?) if FU(x)xF U(x) \to x is an epimorphism. This is equivalent to saying that UU is faithful on morphisms with domain xx. Restricting to the concrete or co-concrete objects are two dual ways to force UU to become faithful.

In the case of local toposes, it is often the concrete objects which we are interested in. When CC is the category of sheaves on a concrete site, then the concrete objects are precisely the concrete sheaves, which form a quasitopos.

On the other hand, in a category of the form scn S(C)scn_S(C), an object (a,x,aU(x))(a,x,a\to U(x)) is concrete just when xx is subterminal. Thus, in this case (such as the category of topological systems), there are very few concrete objects. However, if CC has discrete objects, then (a,x,aU(x))(a,x,a\to U(x)) is co-concrete just when the adjunct map F(a)xF(a) \to x is an epimorphism. Hence the co-concrete topological systems are precisely the topological spaces.

Finally, we can categorify this picture: the notions of discrete and codiscrete objects, fibration, and scone all make sense for higher categories. When we come to concreteness, we have to choose notions with which to categorify “monic” and “epic”. There is probably a reasonable notion of concrete (∞,1)-sheaf in a local (,1)(\infty,1)-topos, though it’s perhaps not immediately obvious what notion of “monic” should be used. And Richard Garner’s ionads are (roughly) the co-concrete objects of the scone of the 2-category ToposTopos, where “epic” is replaced by geometric surjection.

Posted at November 22, 2011 4:53 PM UTC

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10 Comments & 1 Trackback

Re: Discreteness, Concreteness, Fibrations, and Scones

There is probably a reasonable notion of concrete (∞,1)-sheaf

I had forgotten to update that entry and only updated cohesive oo-topos - concrete objects. Now I have copied over some more paragraphs. But there is more to be said…

… though it’s perhaps not immediately obvious what notion of “monic” should be used.

Right, so the updated material speaks about “nn-concrete” objects if XcoDiscXX \to coDisc X is (n-1)-truncated.

Of interest are nn-concrete nn-truncated objects. For instance for AA a concrete sheaf (0-concrete and 0-truncated) the moduli nn-stack B nA\mathbf{B}^n A is nn-concrete and nn-truncated. More interesting is that also its differential refinement B nA conn\mathbf{B}^n A_{conn} is nn-truncated and nn-concrete, but all its (k<n)(k \lt n)-truncations are non-concrete (for any level of concreteness).

More explicitly, in a typical model we have that B nA conn\mathbf{B}^n A_{conn} is presented by a Deligne complex, which is a complex of sheaves that in degree nn is a concrete sheaf and in all lower degrees is a non-concrete sheaf.

Now for Σ\Sigma a rnr \leq n-dimensional manifold, one is interested not quite in the internal hom [Σ,B nA conn][\Sigma, \mathbf{B}^n A_{conn}], but in the (nr)(n-r)-concretefication of its (nr)(n-r)-truncation.

I am supposed to be writing up the story indicated here with Dave Carchedi, but both of us have to find more time….

Posted by: Urs Schreiber on November 22, 2011 10:09 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

Ah, excellent; thank you. Yes, that makes sense. Can you say anything about why you want to truncate and concretify?

Posted by: Mike Shulman on November 22, 2011 10:18 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

Can you say anything about why you want to truncate and concretify?

First, why concretify?

Abstractly, this is related to the fact that the adjunction ismorphism for (Π)(\mathbf{\Pi} \dashv \flat) is not an iso on internal homs: while [ΠX,A][\mathbf{\Pi}X , A] really is the moduli \infty-stack of flat AA-connections on XX, the object [X,A][X, \flat A] is “bigger” in a sense. Only a concretification of it can be equivalent to [ΠX,A][\Pi X, A] (under suitable conditions).

A closely related specific and archetypical example:

consider the sheaf Ω n()Sh(SmthMfd)\Omega^n(-) \in Sh(SmthMfd) of smooth differential nn-forms and let Σ\Sigma be a manifold. Then the internal hom [Σ,Ω n()][\Sigma, \Omega^n(-)] is not quite the “moduli sheaf” of forms on Σ\Sigma. The latter is supposed to send a test manifold UU to the set of UU-parameterized forms on Σ\Sigma. This we can write as the set Ω vert n(Σ×U)\Omega^n_{vert}(\Sigma \times U) of forms that are “vertical” with respect to the projection Σ×UU\Sigma \times U \to U. But

[Σ,Ω n()]:UΩ n(Σ×U) [\Sigma, \Omega^n(-)] : U \mapsto \Omega^n(\Sigma \times U)

is the set of all forms on Σ×U\Sigma \times U. The actual family of forms on Σ\Sigma underlying it is obtained by evaluating on all points of UU, thus killing all forms with “legs along UU”. This is what concretification does.

Secondly, why truncate?

This is an intrinsic way of realizing fiber integrations in differential cohomology. A detailed discussion of the case where one does 0-truncation is here. This gives the higher holonomy. Effectively, for nn-dimensional Σ\Sigma we have that H(Σ,B nA conn)\mathbf{H}(\Sigma, \mathbf{B}^n A_{conn}) is the nn-groupoid of nn-bundles with connection on Σ\Sigma. By dimensional reason they are all flat and trivial. The only characteristic of their equivalence class is their nn-holonomy over Σ\Sigma. So this is what the 0-truncation picks.

Eventually we want to say:

a differentially refined characteristic map

c:BG connB nA conn \mathbf{c} : \mathbf{B} G_{conn} \to \mathbf{B}^n A_{conn}

is an “extended Lagrangian” for a higher Chern-Simons theory, as indicated here, namely an nn-bundle with connection on a moduli \infty-stack of “GG-gauge fields”, where “extended” is in the sense of “extended quantum field theory”, only that here we haven’t quantized yet.

Then for any (kn)(k \leq n)-dimensional manifold Σ\Sigma the transgression of this to the space [Σ,BG conn][\Sigma, \mathbf{B}G_{conn}] of fields on Σ\Sigma (configuration space) is supposed to be the (nk)(n-k)-bundle with connection given by the composite

[Σ,BG conn][Σ,c][Σ,B nA conn]conc nktrunc nk[Σ,B nA conn]B nkA conn. [\Sigma, \mathbf{B} G_{conn}] \stackrel{[\Sigma, \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^n A_{conn}] \to conc_{n-k} trunc_{n-k} [\Sigma, \mathbf{B}^n A_{conn}] \simeq \mathbf{B}^{n-k} A_{conn} \,.

Specifically for k=n1k = n-1 this would be called the prequantum line bundle with connection” on phase space (subject to some technicalities).

So the idea is that extended quantized field theory has an extended prequantum analog, and it works entirely by intrinsic operations in a cohesive context, involving internal homs and their truncation and concretification.

Posted by: Urs Schreiber on November 22, 2011 10:45 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

No time to look closely now, but does your description of ‘ionads’ give us any further clues as to their relationship to modal logic?

Posted by: David Corfield on November 23, 2011 8:51 AM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

does your description of ‘ionads’ give us any further clues as to their relationship to modal logic?

I can’t think offhand of a relationship. There ought to be a relationship between the modality of ionads and the modal logic discussed here, though. Probably Steve Awodey knows what it is.

Posted by: Mike Shulman on December 3, 2011 7:01 AM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

Presumably, in

when the adjunct map F(a)UF(a) \to U is an epimorphism

UU should be xx.

We could do with an nLab entry ‘scone’. Presumably the Sierpinski part of it as to do with Sierpinski space.

Posted by: David Corfield on November 23, 2011 2:36 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

We could do with an nLab entry ‘scone’.

I have started brushing up the topic cluster discrete and concrete objects . Scones still missing, but I need to be doing something else now.

Posted by: Urs Schreiber on November 23, 2011 2:46 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

U should be x.

Thanks, fixed.

Posted by: Mike Shulman on December 3, 2011 7:04 AM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

Cool! Now I know what “concrete” means.

Posted by: Toby Bartels on November 23, 2011 10:09 PM | Permalink | Reply to this

Re: Discreteness, Concreteness, Fibrations, and Scones

(This comment is completely facetious. You have been warned.)

Many poor puns about “concrete scones” spring to mind, linking to Pratchett’s ideas of dwarf bread. But the main objection to the name “scone” is that as an abbreviation of “Sierpinski cone” it has the wrong pronunciation. It’s a “sk-o-n” not a “sk-ow-n”.

Posted by: Andrew Stacey on November 26, 2011 8:49 PM | Permalink | Reply to this
Read the post Scones, Logical Relations, and Parametricity
Weblog: The n-Category Café
Excerpt: The category-theoretic scone or "gluing construction" packages the type-theorist's method of "logical relations" to prove canonicity and parametricity properties of type theory.
Tracked: April 18, 2013 4:55 AM

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