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Stream: theory: category theory

Topic: Inverse image subcategory

Jean-Baptiste Vienney (Aug 14 2024 at 19:24):

This concept seems useful.

Let $F:\mathcal{C} \rightarrow \mathcal{D}$ a functor and let $\mathcal{E}$ be a subcategory of $\mathcal{D}$.

The following defines a subcategory $F^{-1}(\mathcal{E})$ of $\mathcal{C}$:

• $\mathrm{Ob}_{F^{-1}(\mathcal{E})}=\{X \in \mathrm{Ob}_\mathcal{C},~F(X) \in \mathrm{Ob}_\mathcal{E}\}$
• If $A,B \in \mathrm{Ob}_{F^{-1}(\mathcal{E})}$, then $F^{-1}(\mathcal{E})[A,B]=\{u \in \mathcal{C}[A,B],~F(u) \in \mathcal{E}[F(A),F(B)]\}$

The only related thing I find on Google is a notion of "inverse image functor" which is related to sheaves and doesn't seem to be related to this this (but I might be wrong).

Has the inverse image subcategory that I defined above already appeared in the literature? Or have you ever heard about this?

Jean-Baptiste Vienney (Aug 14 2024 at 19:26):

Don't hesitate to tell me also if this notion doesn't make sense. I checked quickly that it works but I may have made a mistake.

Rémy Tuyéras (Aug 14 2024 at 19:34):

Looks like it's a pullback along a monomorphism in $Cat$, hence you get a subcategory on the preimages

Jean-Baptiste Vienney (Aug 14 2024 at 19:34):

Hmm, what is this monomorphism?

Rémy Tuyéras (Aug 14 2024 at 19:35):

Hm, should be something injective on objects and arrows, at least this is the kind of morphisms you are using

Jean-Baptiste Vienney (Aug 14 2024 at 19:36):

I don't get it. I'm talking about a functor without any more assumption.

Rémy Tuyéras (Aug 14 2024 at 19:39):

You have a morphism $\mathcal{E} \hookrightarrow \mathcal{D}$ (excuse my conciseness, I am on my phone)

Jean-Baptiste Vienney (Aug 14 2024 at 19:39):

Oh, ok.

Jean-Baptiste Vienney (Aug 14 2024 at 19:41):

I see, this is the pullback of this inclusion and of $F$.

Rémy Tuyéras (Aug 14 2024 at 19:41):

Yes :)

Jean-Baptiste Vienney (Aug 14 2024 at 19:42):

Cool!

David Egolf (Aug 14 2024 at 19:43):

In case a picture is helpful for anyone:
picture

If $i$ is a monomorphism, then so is $F^*i$. (Pullbacks preserve monomorphisms).

Jean-Baptiste Vienney (Aug 14 2024 at 20:24):

I would be happy to have a bit more.

Suppose I have $\mathcal{E}_2 \le \mathcal{E}_1 \le \mathcal{D}$, I should then have $F^{-1}(\mathcal{E}_2) \le F^{-1}(\mathcal{E}_1)$. So that $F^{-1}$ looks like a functor from the poset of subcategories of $\mathcal{D}$ to the poset of subcategories of $\mathcal{C}$.

Any idea how to phrase this in terms of pullback?

Jean-Baptiste Vienney (Aug 14 2024 at 20:27):

Hmm I think it must be just an application of the universal property!