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Stream: learning: questions

Topic: Accessible two-sided fibration

fosco (Dec 26 2023 at 10:41):

I want to mix two concepts: a two-sided fibration, i.e. a span $A \leftarrow E \to B$ where one leg is an opfibration, the other a fibration, and they are compatible in that opCartesian arrows for one are vertical for the other (and vice versa? I remember one iff the other, but it doesn't really matter now), together with the concept of an accessible fibration as in Makkai-Paré book, definition 5.3.1.

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Given that I tend to reason more often with the associated pseudo(pro)functor $A^o\times B \to Cat$, I would like to know under which conditions it is accessible in the terminology of 5.3.1, so that I can use 5.3.4 (same ref); is this equivalent, or stronger, or weaker, than proving that in the associated span, the legs are respectively an accessible fibration and accessible opfibration?

Mike Shulman (Dec 26 2023 at 18:38):

Note that 5.3.4 applies only to the Grothendieck construction of a contravariant pseudofunctor to produce a fibration. Since the notion of accessible category is not self-dual, this fact doesn't immediately dualize to say anything about the Grothendieck construction of a covariant pseudofunctor to produce an opfibration, let alone the "mixed" Grothendieck construction that produces a two-sided fibration.

Christian Williams (Dec 26 2023 at 19:16):

The concept should be a two-sided fibration internal to Acc, right?

Mike Shulman (Dec 26 2023 at 19:22):

I don't think so, I think it's more subtle than that.

Mike Shulman (Dec 26 2023 at 19:23):

Depending on what you mean by "the concept", of course, but for instance I don't think that definition 5.3.1 is just an ordinary fibration internal to Acc.

fosco (Dec 26 2023 at 22:07):

mh, yes, good point. Ok, then what is missing so that a pseudoprofunctor $A^o\times B \to Cat$ has an accessible total category? Or rather, what would be sufficient?

Mike Shulman (Dec 27 2023 at 08:20):

I don't know, sorry.

Matteo Capucci (he/him) (Dec 27 2023 at 10:48):

I'm inclined to agree with @Christian Williams, what do you think is the subtlety here @Mike Shulman? :thinking:

Matteo Capucci (he/him) (Dec 27 2023 at 13:55):

Uhm now I think I get it, when building the total category of a two-sided fibration one has morphisms from A and B going in opposite directions, roughly meaning colimits in there would be computed 'as colimits' in B and 'as limits' in A, but accessibility only takes care of the former

Mike Shulman (Dec 27 2023 at 16:49):

I'm not necessarily saying it's hard, just that it's different from Makkai-Paré 5.3.4.

fosco (Dec 28 2023 at 09:48):

So, I guess thequestion becomes: if I have this profunctor $P : A^o \times B \to Cat$ and I know that it preserves filtered colimits in both variables separately, and that each reindexing induced by $P(-,b)$ and $P(a,-)$ is accessible (they are left adjoints, if I assume presentability of $A,B$), what can I deduce about the category of elements of $P$?