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Stream: theory: category theory

Topic: Functoriality of categories of presheaves

David Corfield (Jan 31 2025 at 15:32):

On [[functoriality of categories of presheaves]]
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What could we say more precisely about these assumptions on $D$?

David Corfield (Jan 31 2025 at 15:35):

And what could be said on the other side with a functor $D' \to D$ about generating similar adjunctions? What's known about Kan lifts in $Cat$?

Josselin Poiret (Jan 31 2025 at 15:46):

the usual formula for $F_!(P)$ is $F_!(P)(d) = \int^{c:C} \mathrm{Hom}(d, F(c))\cdot P(c)$ where $\cdot$ is the copowering with sets. This only requires that $D$ has all colimits of the size of $C$

Josselin Poiret (Jan 31 2025 at 15:48):

same for $F_*(P)(d) = \int_{c:C}\mathrm{Hom}(F(c), d)\pitchfork P(c)$

Josselin Poiret (Jan 31 2025 at 15:53):

notice that yoneda and coyoneda once again fall from the fact that $\mathrm{Id}^* = \mathrm{Id}$ has unique left and right adjoints, which thus must be the identity

John Baez (Jan 31 2025 at 16:47):

Please, someone, add a remark after this passage explaining some sufficient assumptions! I've bumped into that passage and been annoyed by it. It's good that the detailed assumptions aren't listed there, but it raises curiosity and there should be a remark or link that slakes that curiosity.

David Corfield (Feb 01 2025 at 08:36):

I don't mind adding it, but how should the wording go?

Given some assumptions on $D$ (that it has all limits and colimits the size of $C$), any functor... ?