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Stream: learning: questions

Topic: Name for a kind of subcategories

Jean-Baptiste Vienney (Aug 15 2024 at 15:46):

Is there a name for a subcategory $\mathcal{D}$ of a category $\mathcal{C}$ such that if $f \in \mathcal{D}[A,B]$ is an isomorphism in $\mathcal{C}$, then it is an isomorphism in $\mathcal{D}$ (it is equivalent to requiring that $f^{-1} \in \mathcal{D}[B,A]$)?

Jean-Baptiste Vienney (Aug 15 2024 at 15:47):

Maybe "closed under inverse"?

Nathan Corbyn (Aug 15 2024 at 17:26):

This is the same as asking that the inclusion functor is conservative

Federica Pasqualone (Aug 15 2024 at 19:08):

Well, it can be a replete subcategory, if you mean that: for every $A \in \mathcal{D}_0$ and $f : A \cong B$ in $\mathcal{C}_1$, $f$ and $B$ are in $\mathcal{D}$.

Jean-Baptiste Vienney (Aug 15 2024 at 19:10):

Yeah, I know.

Rémy Tuyéras (Aug 15 2024 at 19:25):

An [[isofibration]] ?

Rémy Tuyéras (Aug 15 2024 at 19:28):

(@Jean-Baptiste Vienney In case this is related to the other post, any kind of fibration defined by lifting property should be stable under pullbacks)

Federica Pasqualone (Aug 15 2024 at 19:36):

Rémy Tuyéras said:

An [[isofibration]] ?

the subcategory is replete iff the inclusion functor is an isofibration :sunglasses:

Jean-Baptiste Vienney (Aug 15 2024 at 19:54):

@Rémy Tuyéras , @Federica Pasqualone this is interesting but slightly different from being closed under inverse.

For instance the full subcategory $\mathcal{D}$ of $\mathbf{FVect}_k$ with objects the vector spaces of the form $k^n$ is closed under inverse but not replete.

Jean-Baptiste Vienney (Aug 15 2024 at 19:59):

And yeah, this is linked to the other post. All this is because I need to be able to work with sub symmetric monoidal categories, direct images of symmetric strict monoidal functors and inverse images of a sub symmetric monoidal category by a symmetric strict monoidal functors. And I need this because I'm working on some characterizations of cartesian symmetric monoidal categories. I try to write a paper with the following abstract:
Screenshot-2024-08-15-at-3.59.20PM.png

Jean-Baptiste Vienney (Aug 15 2024 at 20:00):

I thought it would be 5 pages long but I've already written 50 pages and it's still not done.

Rémy Tuyéras (Aug 15 2024 at 20:07):

Jean-Baptiste Vienney said:

Rémy Tuyéras , Federica Pasqualone this is interesting but slightly different from being closed under inverse.

For instance the full subcategory $\mathcal{D}$ of $\mathbf{FVect}_k$ with objects the vector spaces of the form $k^n$ is closed under inverse but not replete.

Yes, I only read the abstract part of the nlab page and thought it checked out, but the definition indeed diverges from what you want.

It is still a kind of fibration where the inclusion has the right lifting property with respect to the arrow $\{\bullet \to \bullet \} \to \{\bullet \cong \bullet\}$

Rémy Tuyéras (Aug 15 2024 at 20:09):

That right lifting property can give you a factorization system & associated replacement using the small object argument

Jean-Baptiste Vienney (Aug 15 2024 at 20:13):

Brr.. it looks interesting but fibrations are still on the list of stuff I will learn the day I will need it. You have already succeeded to make me understand that I need pullbacks but I still don't crucially need to learn what fibrations are. I'm sure I will need them and appreciate them sooner or later!

Rémy Tuyéras (Aug 15 2024 at 20:22):

OK, no problem. Just in case you are curious about it, you can always go through these very mini exercises to quickly have a good intuition how fibrations are used:
1) Show that a surjection $X \to Y$ is the same as a function that has the right lifting property wrt $\emptyset \to \{\ast\}$
2) Show that an injection $X \to Y$ is the same as a function that has the right lifting property wrt $\{\ast,\ast\} \to \{\ast\}$
3) Show that a full functor $X \to Y$ is the same as a functor that has the right lifting property wrt $\{\bullet, \bullet\} \to \{\bullet \to \bullet\}$
4) Show that a faithful functor $X \to Y$ is the same as a functor that has the right lifting property wrt $\{\bullet \rightrightarrows\bullet\} \to \{\bullet \to \bullet\}$

Exercise Show that if $f:X\to Y$ is a morphism that has the right lifting property wrt to a class of arrows $\mathcal{A}$, then any pullback of $f$ (along any map) also has the right lifting property wrt to $\mathcal{A}$