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Stream: theory: category theory

Topic: epi/mono/? classifiers

Matteo Capucci (he/him) (Apr 06 2020 at 20:50):

Hello folks,
a subobject classifier is basically a mono classifier. Are there any other 'mainstream' notions of classifying objects for a certain kind of morphisms? By this I mean something like "$X$ is a foomorphism classifer if it represents the functor $\mathrm{Foo}(-)$ bringing an object $Y$ to the category of foomorphisms into $Y$ (possibly up to equivalence)".

Paolo Capriotti (Apr 06 2020 at 21:10):

the classifier for principal $G$-bundles?

Joachim Kock (Apr 06 2020 at 21:58):

Matteo Capucci said:

a subobject classifier is basically a mono classifier. Are there any other 'mainstream' notions of classifying objects for a certain kind of morphisms?

The next classifier good to know is the finite-set classifier in the 2-topos of groupoids. (Since John used the word 2-topos in a different way, I should stress that I just mean (2,1)-topos.)

Here is how to find it, and how to see the need of higher toposes. In the category of sets, if you want to classify monos, you should look at the possible fibres of a mono -- they can be empty or singleton -- and then assemble them into the fibres of a family. You get the obvious result: two options, so a base set $\Omega=\{0,1\}$ with the fibre over $0$ empty and the fibre over $1$ singleton.

Now let us try to classify finite sets. There is, up to isomorphism, one finite set for each natural number, so if the classifier exists, it must have the natural numbers as its base, with the fibre over element $n\in \mathbb{N}$ being an $n$-element set, such as $\{0,\ldots,n-1\}$. This nearly works: for any family of finite set (that is, a map $E\to B$ with finite fibres), there is a unique map $\kappa$ from $B$ to $\mathbb{N}$, namely sending $b\in B$ to the cardinality of its fibre. Unfortunately, this is not good enough to be a classifier, because if the fibre has cardinality $n$ then there are $n!$ ways of identifying this fibre with the one of the candidate classifier, $\{0,\ldots,n-1\}$. (Note that this problem does not turn up for the subobject classifier, because sets of size $0$ or $1$ do not have automorphisms.)

The solution is to step up in the categorical hierarchy and keep track of these isomorphisms! So instead of a map of sets having the universal property required, look for a map of groupoids: take as base for the universal family the groupoid $\mathbb{B}$ of all finite set, and take as total space the groupoid $\mathbb{B}'$ of all finite pointed sets. Now for each finite set $S$ (object in the base groupoid), the fibre is precisely that set itself (because there are $S$-many ways of choosing a basepoint in $S$). Now for any family of finite sets ($E\to B$ as before), there is an obvious map $\kappa : B \to \mathbb{B}$, namely sending each $b\in B$ to the fibre $E_b \in \mathbb{B}$. We still have the problem that there are $n!$ ways of identifying such an $n$-element fibre with the corresponding fibre of $\mathbb{B}' \to\mathbb{B}$, but now we are in groupoids, and all these $n!$ identifications are isomorphic! So now it actually works: $\mathbb{B}' \to\mathbb{B}$ is a classifier for finite sets. (The fine print is that the notion of classifier is now in a $2$-categorical sense, but that's the best you can get.)

The moral of the story is that to get a classifier for something $0$-dimensional, it is necessary to step up to the $2$-category of $1$-dimensional objects. (There was nothing special about finite sets; the argument would be the same for $\lambda$-small sets for any regular cardinal $\lambda$.) So this encourages us to work with groupoids instead of sets. Unfortunately, when you start doing this, you will quickly come to a point where you would very much like to have a classifier for groupoids. As you can guess, this is not possible within the setting of groupoids. So you need to step up to $2$-groupoids. And so on. You will never be happy until you come to $\infty$-groupoids. But then your happiness is infinite: you have classifiers for 'all kinds of things' -- general object classifiers. That's the glory of higher topos theory.

(I said 'quickly come to a point'. Historically this step took 40 years. The remarkable step from sets to groupoids in order to find classifiers was found by Grothendieck in the 1960s working in algebraic geometry, needing moduli spaces for all kinds of algebro-geometric objects, such as curves. This led to the notion of stack. Then, in the 1980s, in his 600-page letter to Ronnie Brown and Dan Quillen (and others), he realised that it would be good to have higher stacks. But it is probably fair to say that this dream was only realised in the new millenium.)

Gershom (Apr 06 2020 at 22:14):

In "isotropy and crossed toposes" they construct an internal group object to a topos ($Z$) which "classifies isotropy" (i.e. automorphisms of maps out of objects): http://www.tac.mta.ca/tac/volumes/26/24/26-24.pdf

Mike Shulman (Apr 07 2020 at 02:55):

Back down in 1-category theory, a quasitopos has a classifier for strong monomorphisms. And any extensive category has a classifier for complemented monomorphisms, namely $2=1+1$.

Matteo Capucci (he/him) (Apr 07 2020 at 07:27):

@Joachim Kock your reply is really wonderful! Thanks a lot for the wisdom