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: Coherence of diagrams between presheaf categories

Moana Jubert (Jun 15 2024 at 11:06):

Suppose we have a diagram $F : \mathcal{I} \to \mathbf{Cat}$. It induces a (non-pseudo)functor $\mathrm{PSh}(F) : \mathcal{I}^{\mathrm{op}} \to \mathbf{CAT}$ which sends $i \in \mathcal{I}$ to $\mathrm{PSh}(Fi)$ and arrows $\phi : i \to j$ to $F\phi^\ast : \mathrm{PSh}(Fj) \to \mathrm{PSh}(Fi)$ the "obvious" precomposition.

The general theory of presheaves tells us that each individual $F\phi^\ast$ has a left adjoint, say $L\phi$. Because left adjoints are unique up to isomorphism, composing the left adjoints in $\mathrm{PSh}(F)$ "works" modulo coherence issues, i.e. $L(\phi \circ \psi) \cong L\phi \circ L\psi$.

This is my question: when is it possible to choose left adjoints such that the composition is strict, i.e. get a covariant, (non-pseudo)functor $L : \mathcal{I} \to \mathbf{CAT}$. What are the references?

I would expect the answer depends on the shape of $\mathcal{I}$. Is there an answer when $\mathcal{I} = (\mathbb{N}, \leq)$? I am thinking about the coherence of composing "skeleton functors" between leveled Reedy categories $\mathcal{R}_{\leq 0} \subseteq \mathcal{R}_{\leq 1} \subseteq \mathcal{R}_{\leq 2} \cdots$

Nathanael Arkor (Jun 15 2024 at 11:48):

This is always possible. One way to see this is to observe that the (small) presheaf construction can be strictified into a 2-monad on the 2-category of locally small categories (see the references at [[free strict cocompletion]]) and hence restricts to a 2-functor $\mathbf{Cat} \to \mathbf{CAT}$. Then the assignment you are looking for is simply given by precomposing $F$.

Julius Hamilton (Jun 15 2024 at 12:57):

Suppose we have a diagram $F : \mathcal{I} \to \mathbf{Cat}$.

It induces a (non-pseudo-)functor $\mathrm{PSh}(F) : \mathcal{I}^{\mathrm{op}} \to \mathbf{CAT}$

Is $\mathbf{CAT}$ the category of “large” categories, as opposed to small?

Is there a formal definition of the term “induces”, or does this simply mean “determines”, in that there is a way to construct $\mathrm{PSh}$ uniquely from $F$?

Categorical maps between functors are generally natural transformations; but a natural transformation $\mu : F \to G$ is usually between two functors sharing a source, $F : C \to D$, $G : C \to E$; is that correct?

Is this hinting at a “canonical” natural transformation from say, $F_{small} : \mathcal{I} \to \mathbf{Cat}$ to $F_{large} : \mathcal{I} \to \mathbf{CAT}$, essentially embedding $\mathbf{Cat}$ in $\mathbf{CAT}$, since (I think) $Ob(\mathbf{Cat}) \subset Ob(\mathbf{CAT})$?

The “opposite category” can be defined with a functor. I know natural transformations have a certain commutativity condition. Thus I’m thinking $\mathrm{PSh}$ is ‘induced’ because we can basically map $I$ to $I^{op}$ and $Cat$ to $CAT$ in a unique way.

which sends $i \in \mathcal{I}$ to $\mathrm{PSh}(Fi)$

Moana Jubert (Jun 16 2024 at 08:24):

@Nathanael Arkor Yes, that seems to be what I am looking for. Thank you!

Moana Jubert (Jun 16 2024 at 08:27):

@Julius Hamilton Yes, $\mathbf{CAT}$ is the category of large categories. By "induces" I indeed mean "determines", there is a functor $\mathrm{PSh} : \mathbf{Cat}^{\mathrm{op}} \to \mathbf{CAT}$ which sends a small category to its (large) category of presheaves, and functors to the "obvious" precomposition.

John Baez (Jun 16 2024 at 09:30):

The term "induces" is pretty common when we apply a functor to some map to get a new map.