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Stream: learning: questions

Topic: organizing related cones

David Egolf (Dec 01 2024 at 20:45):

Let $D:J \to [C^{\mathrm{op}}, \mathsf{Set}]$ be a $J$-shaped diagram of presheavevs, where $J$ is a small category. And let us assume that we have a cone under $D$ with tip $X:C^{\mathrm{op}} \to \mathsf{Set}$.

Intuitively, for each $c \in C^{\mathrm{op}}$, we can evaluate our diagram $D$ at $c$, and get a diagram in $\mathsf{Set}$. So we should have some functor $D^t:C^{\mathrm{op}} \to [J, \mathsf{Set}]$. Similarly, by evaluating our cone at any $c$ we get a cone under $D^t(c)$ with tip $X(c)$. So we have a bunch of related cones.

I suspect that we can organize the tips of these cones using a functor $X':C^{\mathrm{op}} \to [J, \mathsf{Set}]$. This functor acts on objects by sending $c$ to the $J$-shaped diagram constant at $X(c)$.

Can we form a natural transformation from $D^t$ to $X'$?

David Egolf (Dec 01 2024 at 20:48):

(For context, if this works out as I hope, I'll be able to finish showing how we can take a colimit of a diagram of presheaves!)

David Egolf (Dec 01 2024 at 22:17):

Perhaps a simpler way to ask this question is this:

We have a bijection $\mathsf{Cat}(J, [C^{\mathrm{op}}, \mathsf{Set}]) \cong \mathsf{Cat}(J \times C^{\mathrm{op}}, \mathsf{Set})$. Can this bijection be upgraded to an isomorphism or equivalence of functor categories?

David Egolf (Dec 01 2024 at 22:21):

If this can be upgraded to an isomorphism or equivalence of categories, then I can hope for the existence of a functor $F:\mathsf{Cat}(J, [C^{\mathrm{op}}, \mathsf{Set}]) \to \mathsf{Cat}(J \times C^{\mathrm{op}}, \mathsf{Set}) \to \mathsf{Cat}(C^{\mathrm{op}} \times J, \mathsf{Set}) \to \mathsf{Cat}(C^{\mathrm{op}}, [J, \mathsf{Set}])$ that is witness to an isomorphism or equivalence.

Nathanael Arkor (Dec 01 2024 at 22:22):

Hint: show that, for any closed monoidal category, there is an isomorphism $[X \otimes Y, Z] \cong [X, [Y, Z]]$, i.e. the "external" tensor-hom adjunction can be represented "internally".

David Egolf (Dec 01 2024 at 22:26):

Thanks for the hint! I was just noticing Proposition 3.1 in this nlab article, which I think covers this point. Maybe I'll give a try at proving the statement you just gave though, before reading the nLab proof!

Peva Blanchard (Dec 01 2024 at 22:29):

A cone under $D$ with tip $X$ can be described as a family of morphisms

$$\iota_{jc} : D(j)(c) \to X(c)$$

It is natural in $c$, meaning that that for every fixed $j$, the sub-family $(\iota_{jc})_c$ is a natural transformation from $D(j)$ to $X$.

It is also natural in $j$, meaning that for every fixed $c$, the sub-family $\mu_c = (\iota_{jc})_j$ is a natural transformation from $D(\_)(c) = D^t(c)$ to the constant functor $X'(c)$ on $J$ with value $X(c)$.

To answer your first question, I think the family $\mu = (\mu_c)_c$ defines a natural transformation from $D^t$ to $X'$.

David Egolf (Dec 01 2024 at 22:32):

Thanks @Peva Blanchard ! Thanks to the comments in this thread, I now have two promising approaches for solving my problem, instead of zero :tada:! I'll plan to work on exploring these approaches tomorrow.

Peva Blanchard (Dec 01 2024 at 22:42):

So the natural transformation amounts to the commutativity of this diagram
image.png

Unfolding the fact these arrows are natural transformations, we obtain the diagrams
image.png

Peva Blanchard (Dec 01 2024 at 22:44):

David Egolf said:

Thanks Peva Blanchard !

you welcome :smile:

David Egolf (Dec 02 2024 at 19:40):

I've made some good progress on understanding this stuff today, but I'm not quite ready to type up here what I have. Hopefully I'll get there soon!