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Stream: learning: questions

Topic: functors compatible with automorphisms

David Egolf (May 22 2023 at 23:56):

Consider the discrete category $2$ with two objects, called $o_1$ and $o_2$. These objects are pretty similar, intuitively, even though they are not isomorphic. In fact, there is an automorphism (an invertible endofunctor) $\alpha: 2 \to 2$ that swaps the two objects, so that in particular $\alpha(o_1) = o_2$.

Now consider a functor $F: 2 \to C$. Then, it is not required by the definition of a functor that $F(o_1) \cong F(o_2)$. I think it is also not required that there exist an automorphism $\beta: C \to C$ so that $\beta(F(o_1)) = F(o_2)$, or even that $\beta(F(o_1)) \cong F(o_2)$.

Sometimes I think of a functor $F: D \to C$ as realizing the structure of $D$ as part of $C$. The situation described above seems to go against this philosophy: $o_1$ and $o_2$ are related in a certain way in $2$, but $F(o_1)$ and $F(o_2)$ are not necessarily related in a similar way in $C$.

Is there a name given to functors that do have the property I discuss above? That is, is there a name for functors $F: D \to C$ so that for any automorphism $\alpha: D \to D$ and any objects $d$ and $d'$ in $D$ so that $\alpha(d) = d'$, there then exists an automorphism $\beta: C \to C$ so that $\beta(F(d)) = F(d')$?

It might be more principled to ask that this only hold up to isomorphism. So, I would also be interested in considering a relaxed version of this condition: Is there a name for functors $F: D \to C$ so that for any automorphism $\alpha: D \to D$ and any objects $d$ and $d'$ in $D$ so that $\alpha(d) \cong d'$, there then exists an automorphism $\beta: C \to C$ so that $\beta(F(d)) \cong F(d')$?

(A follow-up thought: Maybe things would be nicer if I didn't consider automorphisms of categories, but instead equivalences of a category with itself.)

Kevin Arlin (May 23 2023 at 00:19):

I don't think this is really about categories, actually. Groups, for instance, have the same behavior you complain about, where the existence of an automorphism of $C$ sending $g_1$ to $g_2,$ which certainly says there's something similar about $g_1$ and $g_2$ in $C,$ doesn't imply the same for $F(g_1),F(g_2)$ when $F:C\to D$ is a homomorphism. (Note that I'm using your notation but there's a level shift here, if we think about groups as categories: $g_1,g_2$ are morphisms.)

Whether in groups or categories, I'm not really sure how to think about your proposed condition. It seems awkward to work just one object at a time. Maybe it would be more natural (without messing up your motivating example) to ask that for any automorphism $\alpha$ there exist $\beta$ such that $\beta\circ F=F\circ\alpha$? It's hard for me to see what this property means, though, except in the case that $F$ is an isomorphism or an equivalence, where it's always possible by setting $\alpha=F^{-1}\beta F.$

Morgan Rogers (he/him) (May 23 2023 at 08:20):

This kind of property is more commonly considered as a property of the codomain rather than the functor. There are concepts in model theory of homogeneous object (relative to some class of objects). The "random graph" is an example of this: given any finite graph and two embeddings of that graph into the random graph (or equivalently, given two isomorphic subgraphs), there is an automorphism of the random graph exchanging them.

Morgan Rogers (he/him) (May 23 2023 at 08:21):

Actually, maybe "ultrahomogeneous" is the term I should be using for this property. Here's a MO question about those.

Morgan Rogers (he/him) (May 23 2023 at 10:14):

Incidentally, a related property studied by Ehresmann and Vanbremeersch (leading to their definition of "multiplicity principle") appears in a paper I am currently reviewing. Hopefully that gives you some keywords to search for.

David Egolf (May 23 2023 at 15:03):

Thanks to both of your for your interesting responses! After thinking about them for a bit, I feel like I understand what's going on here a little better. I appreciate the reference to the concept of "ultrahomogeneous" in model theory.

I think it makes sense to me that this kind of property is more often considered as a property of the codomain. I suppose it's actually quite a lot to ask that automorphisms $\alpha$ of $D$ extend to automorphisms $\beta$ of $C$ for $F: D \to C$. Intuitively, for some object $d \in D$, $F(d)$ and $(F \circ \alpha)(d)$ might be connected to other parts of $C$ in different ways, which starts adding in extra conditions (relating to the structure of $C$) for any automorphism of $C$ that sends $F(d)$ to $(F \circ \alpha)(d)$.

By the way, this question occurred to me while I was thinking about limits. I was wondering if $\lim F$ and $\lim (F \circ \alpha)$ are necessarily isomorphic if they both exist, for $\alpha$ an invertible endofunctor. I am still hoping this might be true!

Josselin Poiret (May 23 2023 at 19:31):

yes, in general any equivalence $\alpha$ would give you an isomorphism of limits