John wrote
(confessionally)
My brain still tends to freeze over when my eyes see the word
‘classifying topos’
and (helpfully)
Classifying topoi are a bit tricky for me. They’re just like Lawvere’s
‘algebraic theories’, but instead of describing some sort of
mathematical gadget using a category with finite products, we use a
full-fledged topos. So, for example, the classifying topos for groups
is ‘the free topos on a group object’, just as the algebraic theory
for groups is ‘the free category with finite products on a group
object’. It’s just a more elaborate version of the same idea.
and (hopefully)
I only mentioned them to lure Todd and other hotshots into posting
some comments on this thread.
While I’m certainly not a topos hotshot, and it’s clear that the right
person to explain Topos Theory is someone with initials TT, I’ll have
a go at explaining classifying toposes.
Preamble
First, I agree with the paragraph of John’s quoted above. Saying that
some topos is the classifying topos for Boolean algebras is very
similar in spirit to saying that some finite product category is the
Lawvere theory of groups, or saying that the monoidal category of
1-manifolds and cobordisms is the free monoidal category on a
commutative Frobenius algebra.
Certain kinds of theory (e.g. the theory of monoids) can be
interpreted in a monoidal category. In other words, if $\mathcal{E}$
is a monoidal category, it makes sense to talk about models of such a
theory in $\mathcal{E}$. For more general kinds of theory you have
to restrict to finite product categories. For more general kinds
still, you have to restrict to toposes.
However, when you get to the topos level, there are two small
wrinkles. In a way I don’t want to mention them, because they don’t really matter for the explanation I’m going to give. On the other
hand, they were a source of confusion to me when I first learned this
stuff, so I’ll go ahead. Skip to the definition of classifying topos if
you like.
One wrinkle concerns maps of toposes. If you take the definition
of topos (a cartesian closed category with finite limits and a
subobject classifier) and write down what seems to be the obvious
notion of a map of toposes (a functor that preserves all this
structure, up to isomorphism), that’s not actually a very useful
notion. It turns out that the best notion of a map of toposes is
that of geometric morphism: an adjunction with a certain
property (that the left adjoint preserves finite limits). I can’t
give a good conceptual explanation for this. A practical
explanation is that a continuous map $X \to Y$ of topological spaces
induces a geometric morphism $Sh(X) \to Sh(Y)$ between toposes of
sheaves, but doesn’t in general induce a map of toposes in the
‘obvious’ sense.
This brings us to the other wrinkle. Because a map of toposes is an
adjunction, we have a choice of orientation. The custom is to write
geometric morphisms in the direction of the right adjoint. As we’ve
just seen, this convention fits with the view of toposes as
generalized spaces. But — perhaps because of the usual
geometry/algebra duality? — it also means that when we come to
do classifying toposes of algebraic theories, things go
‘back-to-front’. So while the Lawvere theory of groups is the finite
product category $\mathbf{T}$ with the property that for any finite
product category $\mathcal{E}$, groups in $\mathcal{E}$ are the same
as finite-product-preserving functors
$\mathbf{T} \to \mathcal{E},$
the classifying topos for groups is the topos $\mathbf{U}$
with the property that for any topos $\mathcal{E}$, groups in
$\mathcal{E}$ are the same as geometric morphisms
$\mathcal{E} \to \mathbf{U}.$
What presheaves classify
The simplest toposes are the presheaf toposes. For a
small category $\mathbf{C}$, the presheaf topos on $\mathbf{C}$ is by
definition the category $[\mathbf{C}^op, Set]$ of contravariant
set-valued functors on $\mathbf{C}$. Simplicial sets (where
$\mathbf{C} = \Delta$) and symmetric sets (where $\mathbf{C} =
FinSet$) are examples.
So, what do presheaf toposes classify? In other words, given a topos
$\mathcal{E}$, what’s a geometric morphism
$\mathcal{E} \to [\mathbf{C}^op, Set] ?$
Of course, it could be that it just is what it is. But in fact
there’s a nice theorem: a geometric morphism of this form
is the same thing as a flat functor $\mathbf{C} \to \mathcal{E}$. So, $[\mathbf{C}^op, Set]$ classifies flat functors out of $\mathbf{C}$.
OK… but what’s a flat functor? I won’t give the definition, but to
a first approximation, a flat functor is a functor that preserves
finite limits. This is actually true when $\mathbf{C}$ has
finite limits. It’s also true that flat functors preserve all
finite limits that exist in $\mathbf{C}$.
It’s somehow a bit daft to think about finite-limit-preserving functors on a category that doesn’t have all finite limits; flatness is the righteous concept.
Let’s look at the two presheaf toposes that John mentioned: simplicial
and symmetric sets.
By the theorem above, $[\Delta^op, Set]$ classifies flat functors out
of $\Delta$. Since $\Delta$ doesn’t have all finite limits (e.g. it
doesn’t have binary products), it’s not terribly easy to see what a
flat functor out of $\Delta$ is.
Similarly, the topos $[FinSet^op, Set]$ of symmetric sets classifies
flat functors out of $FinSet$. This time we’re in luck: $FinSet$ has
finite limits, so flat $=$ finite-limit-preserving. But it’s not so
easy to see what a finite-limit-preserving functor out of $FinSet$ is,
either!
So it would be good to have some heavier artillery, and that’s what we
come to next.
The classifying topos of a finitary algebraic theory
Finitary algebraic theories can be interpreted in any finite product
category: for instance, you can talk about a group in any finite
product category. Toposes are rather special finite product
categories, so we can interpret a rather wider class of theories in
them (the ‘geometric theories’). But in particular, we can still
interpret finitary algebraic theories.
We can therefore ask: what is the classifying topos of a finitary
algebraic theory? For instance — taking the standard example
— what topos $\mathbf{G}$ has the property that for any topos
$\mathcal{E}$, geometric morphisms
$\mathcal{E} \to \mathbf{G}$
are the same thing as groups in $\mathcal{E}$?
The answer turns out to be wonderfully simple:
The classifying topos of the theory of groups is $[Gp_fp, Set]$,
where $Gp_fp$ is the category of finitely presentable groups
… and the same is true if you replace ‘the theory of groups’ by any other finitary
algebraic theory.
Example The simplest of all algebraic theories is the
theory of sets. This is the theory with no operations and no
equations. A true categorical logician sees nothing special about
taking models in the category of sets (boring!), so would call
this ‘the theory of objects’.
A finitely presentable set is simply a finite set. So the classifying
topos for the theory of objects is $[FinSet, Set]$. In other words,
for any topos $\mathcal{E}$, a geometric morphism
$\mathcal{E} \to [FinSet, Set]$
is the same thing as an object of $\mathcal{E}$.
These are covariant functors on $FinSet$, so they’re not
symmetric sets.
Example The classifying topos for the theory of Boolean
algebras is $[Bool_fp, Set]$, where $Bool_fp$ is the category of
finitely presented Boolean algebras.
Stone duality says that the category of Boolean algebras is dual to
the category of Stone spaces ($=$ compact Hausdorff totally
disconnected spaces). Finite presentability of a Boolean algebra is
equivalent to finiteness of the corresponding space. But a finite
Stone space is just a set, so $Bool_fp \simeq FinSet^op$. Hence (aha!)
the classifying topos for Boolean algebras is $[FinSet^op, Set]$,
symmetric sets.
Example This one’s a bit of a cheat, because we’re going
to consider the theory of totally ordered sets with top and bottom
(intervals, let’s say), and that’s not quite an algebraic theory.
Regardless, the theory of intervals does have a classifying
topos, namely $[Intvl_f, Set]$, where $Intvl_f$ is the category of
finite intervals.
A nice little duality says that $Intvl_f$ is dual to the category
$\mathbf{D}$ of finite totally ordered sets. (Hint: think about
homming into the two-element totally ordered set.) So intervals are
classified by the topos $[\mathbf{D}^op, Set]$ of augmented simplicial
sets.
Example If you want actual simplicial sets — that
is, presheaves on the category $\Delta$ of nonempty finite
totally ordered sets — then you can take the theory of
nontrivial intervals, i.e. those in which the top and bottom
are distinct. Thus, the topos of simplicial sets classifies the theory of strict
intervals.
Similarly, presheaves on the category of nonempty finite sets
classify nontrivial Boolean algebras.
Help, Todd
One thing puzzles me. According to p.2 of the paper of
Lawvere mentioned by Charles Stromeyer, Boolean algebras are
classified by presheaves on the category of nonempty finite sets. But
according to what I’ve written above, they’re classified by presheaves
on the category of all finite sets (which isn’t Morita equivalent).
Either one of us is wrong (and I don’t fancy my chances against
Lawvere on topos theory) or Lawvere takes Boolean algebras to be
nontrivial by definition (which seems unlikely). Help!
Re: Geometric Representation Theory (Lecture 2)
I hope some people, at least Todd, understand what Jim is up to in this seminar. I feel a bit sad sometimes when we get lots of nice comments on Jeremy Bentham’s head or old Moody Blues songs, but none about a piece of mathematics that’s really quite profound. Of course it’s a lot of work to watch a 1-hour video of a math talk and then figure out that beneath the gentle surface there’s some heavy-duty math that really deserves more explication…
Anyway, something like this seems to be going on. Given a finite set $S$ and a finite group $G$ acting on it, we want to completely describe the action in terms of the invariant $n$-ary predicates on $S$. These are the same as the invariant subsets of $S^n$, and these are just subsets of $S^n/G$.
So, to each $n$ we should keep track of $S^n/G$. But, if we think of $n$ as a finite set rather than just a natural number, we have
$S^n/G = hom(n,S)/G$
Then we see that any function
$f: n \to m$
induces a function from $hom(m,S)/G$ to $hom(n,S)/G$, which deserves to be called
$hom(f,S)/G : hom(m,S)/G \to hom(n,S)/G$
The extra information in these functions is very important since it says how to use a function $f: n \to m$ to ‘substitute variables’ in an invariant $n$-ary predicate to get an invariant $m$-ary predicate. (Yes, I think there’s an extra contravariant twist here.)
This is where Lawvere’s work on ‘existential and universal quantifiers as adjoints to substitution’ comes in, but never mind… What we get is a functor
$hom(-,S)/G : FinSet^{op} \to Set$
Now, a functor from $FinSet^{op}$ to $Set$ is called a symmetric set — it’s a close relative of a simplicial set, where instead of $FinSet$ we use $\Delta$, the category of totally ordered finite sets.
So, we can try to draw a symmetric set as something a bit like a simplicial set, but where the edges of our simplices don’t point in a specific direction.
If we draw the symmetric set $hom(-,S)/G$ as something like a simplicial set, and don’t bother drawing the ‘degeneracies’, and then take its barycentric subdivision, we get Jim’s orbi-simplex. I think. (This is all stuff he explained to me while he was developing the idea.)
What’s really cool is the relation between the orbi-simplex and Young diagrams. This deserves to be worked out quite formally.
To add to the fun, various category theorist have already studied symmetric sets — and this is where Todd might be able to help me out. I seem to recall that the category of symmetric sets is the classifying topos for Boolean algebras, or something like that. And this should have something to do with the ‘logic-flavored’ aspect of what we’re doing here: studying a Boolean algebra of $G$-invariant predicates on a set $S$. But, I don’t exactly see the connection. My brain still tends to freeze over when my eyes see the word ‘classifying topos’ — that’s part of the problem.
There’s also other work on symmetric sets, which may or may not be relevant.