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Stream: learning: questions

Topic: Subobject classifier in ∞Grpd

Jonathan Arnoult (Oct 08 2024 at 16:11):

A simple question but, ∞Grpd is a (∞,1)-topos, so in particular has a subobject classifier. What should it be? I only know the example of Set being a topos where the subobject classifier is the boolean set and the true morphism from 1 to that set, and can't see really how to generalize this. I thought it could be the discrete category with two 0-cells, and zero non-trivial n-cells otherwise, and the true functor just picking one of those two 0-cells, but that's just an intuition from the case of Set. And I can't find anywhere something unfolding explicitely the topos properties in the case of ∞Grpd.

Morgan Rogers (he/him) (Oct 08 2024 at 16:43):

That sounds like a good guess to me. Don't the infinity topos resources that you've read mention even the simplest case when they talk about subobject classifiers?

Chris Grossack (they/them) (Oct 08 2024 at 16:50):

In general, an $(n+1)$-topos has $n$-truncated spaces as its objects, and has an $(n-1)$-object classifier which is given by the "forgetful" map from pointed objects to unpointed ones.

For instance, $\mathsf{Set}$ is a $1$-topos whose objects are $0$-truncated, and it has a $-1$-object classifier. Recall the "$-1$-truncated objects" are propositions, so maps $X \to \Omega$ classify "proposition bundles" over $X$. But (thinking classically for a second for familiarity) a proposition is either true or false, so a "proposition bundle" chooses either true or false over each element of $X$, and this exactly names a subobject of $X$. Also, note that in this case the universal family $1 \overset{\top}{\to} \Omega$ is given by the forgetful map from "pointed propositions" $1$ to "all propositions" $\Omega$.

Chris Grossack (they/them) (Oct 08 2024 at 16:54):

Similarly, if you look at the 2-topos of 1-groupoids, this will have a 0-object classifier. That is, an object (which I'll still call $\Omega$) so that maps $X \to \Omega$ classify "set bundles" over $X$. In more familiar words, covering spaces!

Again, we can compute exactly what this classifier is, it's the forgetful map from {pointed sets} to {sets}, and it's a fun exercise to convince yourself that this really does classify set-bundles/covering spaces on groupoids

Chris Grossack (they/them) (Oct 08 2024 at 16:55):

And, of course, if you look at the $\infty$-topos of $\infty$-gropuoids, there's an $\infty$-object classifier, which is the usual univalent universe from HoTT. If you've looked into the HoTT semantics at all, you might remember the universe is given by the forgetful map from pointed objects to objects, which matches the rest of the pattern! Maps to this object classify all bundles over an object, which makes the theory especially pleasant! Informally this is because $\infty+1 = \infty = \infty-1$, so everything exists at the same level.

Chris Grossack (they/them) (Oct 08 2024 at 16:55):

You can read more (though, unfortunately, not much more) at the nlab here and here

John Baez (Oct 08 2024 at 17:01):

I think Chris is saying it pays to revise our concept of 'subobject' as we march up the ladder of dimensions. For a 0-groupoid or set, a subobject tells us which elements have a certain property. For a 1-groupoid, we may instead be interested in which objects have a particular structure. (We could also be interested in a property: this is a special case worthy of study, but just a special case.) For a 2-groupoid, we may instead by interested in which objects are equipped with some particular stuff. And so on. Our naive intuitions about subobjects need to be revised, if we want to go down this road.

Chris Grossack (they/them) (Oct 08 2024 at 18:00):

Oh also, I realize now that I misread your question. The subobject classifier in $\infty\text{Gpd}$ is exactly the same as in $\mathsf{Set}$. It's the map $\top : \{ \star \} \to \{ \top, \bot \}$, but now we view these as discrete sets.

Daniel Gratzer (Oct 08 2024 at 18:06):

Just to give another reference, what @Chris Grossack (they/them) mentioned is Example 6.1.6.2 in higher topos theory. All of 6.1.6 deals with classifying maps in topoi and is quite nice

Jonathan Arnoult (Oct 08 2024 at 18:39):

Thanks for your answers!

Daniel Gratzer (Oct 08 2024 at 19:17):

Just as a mild follow up: every $\infty$-topos $\mathcal{E}$ has an associated 1-topos of its 0-truncated objects $\mathcal{E}_{\le 1}$. The subobject classifier of $\mathcal{E}$ is always given by the subobject classifier $\Omega$ of $\mathcal{E}_{\le 1}$.

I haven't seen this written down before categorically (if you're familiar with HoTT, this is the claim that $\mathrm{hProp}$ is an h-set), so here's my quick proof sketch: since $\Omega$ is 0-truncated, we know that $\hom(X, \Omega) \cong \hom(\tau_{\le 0} X, \Omega) \cong \mathsf{Sub}(\tau_{\le 0} X)$. It therefore suffices to show that the map $\mathsf{Sub}(\tau_{\le 0} X) \to \mathsf{Sub}(X)$ is bijective. It is injective because $X \to \tau_{\le 0} X$ is an effective epimorphism and surjective because the same map is in fact 0-connected. This should all work replacing subobject classifiers with object classifiers for higher truncation levels.

Joe Moeller (Oct 08 2024 at 20:51):

Chris Grossack (they/them) said:

Oh also, I realize now that I misread your question. The subobject classifier in $\infty\text{Gpd}$ is exactly the same as in $\mathsf{Set}$. It's the map $\top : \{ \star \} \to \{ \top, \bot \}$, but now we view these as discrete sets.

This confuses me. For example, I would guess that this isn't enough to see subgroups, thought of as one-object groupoids, thought of as $\infty$-groupoids.

Daniel Gratzer (Oct 08 2024 at 20:58):

Joe Moeller said:

Chris Grossack (they/them) said:

Oh also, I realize now that I misread your question. The subobject classifier in $\infty\text{Gpd}$ is exactly the same as in $\mathsf{Set}$. It's the map $\top : \{ \star \} \to \{ \top, \bot \}$, but now we view these as discrete sets.

This confuses me. For example, I would guess that this isn't enough to see subgroups, thought of as one-object groupoids, thought of as $\infty$-groupoids.

The trick is that 'monomorphisms' in higher category theory are often more restrictive than one might guess. In this case, a monomorphism is a (-1)-truncated map of spaces: every (homotopy) fiber is either empty or contractible. The result is that the only monomorphisms are the inclusions of connected components into a space.

Joe Moeller (Oct 08 2024 at 21:01):

Ok right, I thought the two options were that ordinary subgroups aren't allowed to be subobjects here because somehow subobjects can only be connected components, or that there was some sneaky extra data hiding in the upgraded version of this subobject classifier so that somehow subgroups squeeze in anyway.

Kevin Carlson (Oct 08 2024 at 21:44):

I always find it hopeless to navigate the maze of articles on the nLab around $n$-monomorphisms, truncated morphisms, and suchlike, which all seem to refer to each other in a loop that never quite gets to what I actually want to know, but I think that a $2$-monomorphism in an $\infty$-category is a map that's mono on $\pi_0,\pi_1$, more or less, which gets your subgroups. And I'm pretty sure there are classifiers for $n$-monomorphisms for arbitrary $n$ in an $\infty$-topos.

Kevin Carlson (Oct 08 2024 at 21:45):

OK, yeah, this page backs up that the $0$-truncated morphisms between groupoids are the faithful functors, and (uhh) I think there's some $+2$ shift between the indexing for kinds of mono and for kinds of truncation. https://ncatlab.org/nlab/show/%28n-connected%2C+n-truncated%29+factorization+system

Daniel Gratzer (Oct 09 2024 at 12:01):

Kevin Carlson said:

I always find it hopeless to navigate the maze of articles on the nLab around $n$-monomorphisms, truncated morphisms, and suchlike, which all seem to refer to each other in a loop that never quite gets to what I actually want to know, but I think that a $2$-monomorphism in an $\infty$-category is a map that's mono on $\pi_0,\pi_1$, more or less, which gets your subgroups. And I'm pretty sure there are classifiers for $n$-monomorphisms for arbitrary $n$ in an $\infty$-topos.

It shouldn't need to a be a monomorphism on $\pi_0$ I think. One can usually work this out using the long exact sequence of homotopy groups, since the condition of $f : X \to Y$ being n-truncated amounts to its fibers being n-truncated spaces (vanishing homotopy groups above n). So 0-truncated maps are injective on $\pi_1$ and bijective above it and, more generally, n-truncated maps should be injective on $\pi_n$ and bijective above it.

This means (somewhat counter-intuitively) that $\mathbf{2} \to \mathbf{1}$ is a 0-truncated map. For this reason, I think it's less confusing to stick to truncated and avoid 0-monomorphism.

I do agree that there is a classifier for ($\kappa$-small) n-truncated maps in any m-topos whenever $m \ge 1 + n$.

Madeleine Birchfield (Oct 09 2024 at 16:10):

Doesn't the existence of object classifiers in $\infty\mathrm{Grpd}$ depend on the existence of at least one inaccessible large cardinal? Like I don't think the category of $\infty$-groupoids associated with the initial model of ZFC or ETCS has an object classifier.

Daniel Gratzer (Oct 09 2024 at 16:36):

Madeleine Birchfield said:

Doesn't the existence of object classifiers in $\infty\mathrm{Grpd}$ depend on the existence of at least one inaccessible large cardinal? Like I don't think the category of $\infty$-groupoids associated with the initial model of ZFC or ETCS has an object classifier.

Indeed, and to be precise one has object classifiers associated to each sufficiently large regular cardinal in the ambient set theory. Unlike subobject classifiers, there is no single object classifier of (say) (n+1)-truncated morphisms: one has to talk about a classifier for fiberwise $\kappa$-compact (n+1)-truncated morphisms.

Edit: Whoops: as mike points out, the object classifiers are _sensitive_ to what cardinals you have around, but you don't need them to be inaccessible

Mike Shulman (Oct 09 2024 at 17:01):

It depends on what you want your object classifiers to be closed under. There is a classifier of fiberwise $\kappa$-compact $n$-truncated morphisms for any regular cardinal $\kappa$, and there are plenty of regular cardinals in ZFC without any inaccessibles. It's just that the maps classified by such a classifier are not closed under all constructions like $\Pi$-types unless $\kappa$ is inaccessible.