Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Slice Categories and the Hom(-, A) Functor

Sean Gloumeau (Oct 16 2023 at 14:02):

I've been thinking about parallels and differences between the slice category over A $\mathbf{C}/A$ and the objects and morphisms in $\mathbf{Set}$ mapped to by the Hom(-, A) functor. It seems like we can define a functor $F: \mathbf{C}/A \rightarrow \mathbf{Set}$ to get something like a natural ismorphism defined by inclusion of a morphism into its hom-set that sends $f: B \rightarrow A$ to $hom(B, A)$ and commutative diagram $h = fg$ to the pullback $g^*$. Assuming the hom functor and $F$ are from the same categories (which they aren't, for the record), a natural isomorphism could be constructed by taking $\eta_X = id_X$ for all hom-sets $X \in \mathbf{Set}$ mapped to by the functors.

Anyway, I'm curious to hear if there's a nicer category-theoretic way to think of the similarities between $\mathbf{C}/A$ and Hom(-, A), and situations where thinking about morphisms in a category using one or the other is more useful

Todd Trimble (Oct 16 2023 at 14:38):

There is a famous equivalence between the slice of presheaves $\mathbf{Set}^{C^{op}}/\hom(-, A)$ and presheaves on the slice, $\mathbf{Set}^{(C/A)^{op}}$.

Sean Gloumeau (Oct 16 2023 at 14:52):

Oh this'll be fun to play around with and see if I can find and make sense of the equivalence myself, thanks for putting me on :)

Patrick Nicodemus (Oct 16 2023 at 15:14):

A reference for this is in the book by Kashiwara and Schapira, "Categories and sheaves". I don't have a copy on hand so i can't tell you where.