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Stream: learning: questions

Topic: "re-strictification" of pseudofunctors

Tim Hosgood (Jul 17 2023 at 12:39):

Given a pseudofunctor $F\colon\mathcal{C}\to\mathsf{Cat}$ there is the notion of strictification (or sometimes rectification), which gives a 1-functor $\widehat{F}\colon\mathcal{C}\to\mathsf{Cat}$ . One key example of a pseudofunctor (for me) arises from sheaves of $\mathcal{O}_X$ -modules: given morphisms $(X,\mathcal{O}_X)\xrightarrow{f}(Y,\mathcal{O}_Y)\xrightarrow{g}(Z,\mathcal{O}_Z)$ of ringed spaces, the pullback $f^*=f^{-1}(-)\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$ forms part of a pseudofunctor (that sends a ringed space to its category of modules), since $(gf)^*\cong f^*g^*$ holds only up to natural isomorphism.

But in this specific case, I think you can avoid strictification by instead restricting (whence the weak pun in the title) your pseudofunctor down to some smaller category (like the wide subcategory of ringed spaces where we only consider open immersions, since then the pullback is "just" restriction) and obtaining a 1-functor in this way.

Is this specific example true? I get confused when I think about the fact that open immersions aren't just inclusions of open subsets, but can possibly also contain some non-trivial isomorphism.
Is there a general theory/name for this, where you strictify a pseudofunctor by restricting to a particularly nice subcategory (I'd like to say "the maximal wide subcategory such that the pseudofunctor is a functor")? If so, is this somehow dual to the usual notion of strictification/rectification?

Tim Hosgood (Jul 17 2023 at 19:27):

ok, so I think the answer to question 1 is a bit too much algebraic geometry to be really on topic for this zulip (in hindsight, I was really only thinking of the baby version where you work with the category of open sets in some fixed space, and the sheaves really are sheaves of functions)

Tim Hosgood (Jul 17 2023 at 19:27):

but i'm still interested in question 2!

Mike Shulman (Jul 17 2023 at 19:31):

I don't see how it could work in general, because strictness of a pseudofunctor is detected at pairs of morphisms rather than single ones, so you couldn't say something like "consider all the morphisms on which F is strict".

Tim Hosgood (Jul 17 2023 at 19:35):

that's very true, I guess in the toy example I was thinking about there happens to be a "type" of morphism that is strict enough such that the property is guaranteed on all pairs (namely "inclusion of an open subset")

Tim Hosgood (Jul 17 2023 at 19:35):

maybe this is just very special to that one case though

Mike Shulman (Jul 17 2023 at 19:39):

There are other examples like that, but the "type of morphism" in general is extra data, not something that can be universally reconstructed from the pseudofunctor. You can talk about it in the abstract context of [[double categories]], [[proarrow equipments]], [[F-categories]], etc.

Tim Hosgood (Jul 17 2023 at 20:19):

do you think you could say a little bit more about how e.g. double categories might be useful in talking about this example (or similar)?

Mike Shulman (Jul 19 2023 at 18:53):

If you have a category $C$ containing a wide subcategory $A$ of "nice" morphisms, you can make it a double category with $A$ as the tight morphisms and $C$ as the loose morphisms. Call this $Q(C,A)$ . Then if $K$ is a 2-category and you make it into a double category $Q(K)$ of [[quintets]], a "pseudo double functor" (which usually means pseudo on the loose morphisms and strict on the tight ones) $Q(C,A) \to Q(K)$ is essentially a pseudofunctor $C\to K$ together with a strictification of its restriction to $A$ .

Tim Hosgood (Jul 19 2023 at 20:17):

oh that's so neat!