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: theory: category theory

Topic: minimal and maximal objects

Matteo Capucci (he/him) (Mar 11 2023 at 22:30):

In preordered sets it is customary to speak of minimal and maximal elements, i.e. elements $a \in A$ such that if $b \leq a$ (resp. $A \leq b$) then $b = a$. These are notoriously different from minima and maxima, which in category theory correspond to initial and terminal objects in a category. I've never heard of minimal or maximal objects though. Any idea why?

Matteo Capucci (he/him) (Mar 11 2023 at 22:34):

Here's a working definition:

A minimal object in $\cal C$ is an object $a : \cal C$ such that any morphism $b \to a$ is invertible

Matteo Capucci (he/him) (Mar 11 2023 at 23:00):

To cite something that already exists, a [[strict initial object]] would then be a minimal initial object

David Egolf (Mar 12 2023 at 01:47):

Thinking about this, I was wondering if it could be interesting to talk about "local" limits or colimits in general. Call a subcategory $\mathsf{B}$ of $\mathsf{C}$ "full at $a$" if $\mathsf{B}(a,b) = \mathsf{C}(a,b)$ and $\mathsf{B}(b,a) = \mathsf{C}(b,a)$ for all objects $b$ in $\mathsf{C}$. The idea is to keep in $\mathsf{B}$ all the morphisms to or from $a$. Then one could say that $a$ is "locally" some limit or colimit if it is that limit or colimit in some subcategory that is full at $a$.

Then maybe a minimal object in the sense @Matteo Capucci (he/him) describes above is an object that is "locally" an initial object? And a maximal object would be one that is "locally" a terminal object? (I haven't thought about this carefully, though. Hopefully I didn't confuse things.)

EDIT: No, these concepts are different. Consider a category with two objects $A$ and $B$ and two morphisms $f$ and $g$ from $A$ to $B$. Then $A$ is minimal but it is not locally an initial object in the sense I describe above.

Nathanael Arkor (Mar 12 2023 at 16:06):

Just to remark that this question was recently asked on Math.SE (with the same proposed definition).

Dylan McDermott (Mar 12 2023 at 16:15):

There is a notion of minimal object in Definition 4.1 of https://arxiv.org/abs/2202.05701

Morgan Rogers (he/him) (Mar 12 2023 at 16:16):

Without imposing further properties, it's interesting to note that any endomorphisms of a minimal or maximal object is forced to be invertible (but that the unique object of the unlooping of a group has this property), and that various common structural conditions on categories force such objects to be unique if they exist. For instance, a minimal object in a category with products is unique and weakly initial (admits at least one morphism to every object).

Matteo Capucci (he/him) (Mar 12 2023 at 16:59):

Interesting @Morgan Rogers (he/him)

Matteo Capucci (he/him) (Mar 12 2023 at 17:00):

Why does a minimal object in a category with products have to be unique?

Matteo Capucci (he/him) (Mar 12 2023 at 17:00):

Ah I see. Let $a,b$ both be minimal. Then the projections $a \times b \to a, b$ are isomorphisms, implying both are isomorphic to $a \times b$.

Matteo Capucci (he/him) (Mar 12 2023 at 17:01):

Also $a \times a \cong a$ and $a \times b \cong a$ for any $b$!

Matteo Capucci (he/him) (Mar 12 2023 at 17:01):

This reminds me of a [[reflexive object]]

Matteo Capucci (he/him) (Mar 12 2023 at 17:06):

Dylan McDermott said:

There is a notion of minimal object in Definition 4.1 of https://arxiv.org/abs/2202.05701

This seems interesting, thanks for the pointer

Matteo Capucci (he/him) (Mar 13 2023 at 10:10):

@Dylan McDermott's suggestion was spot on, the paper gives a very compelling definition of minimality. So I started an nLab entry about it: [[minimal object]]

Hugo Bacard (Mar 13 2023 at 13:41):

Matteo Capucci (he/him) said:

Here's a working definition:

A minimal object in $\cal C$ is an object $a : \cal C$ such that any morphism $b \to a$ is invertible

I might be wrong but is it equivalent to saying that the comma category $(\cal C \downarrow a)$ is a groupoid ?

John Baez (Mar 13 2023 at 15:33):

A morphism in $\mathcal{C}\downarrow a$ from $f: b \to a$ to $f' : b' \to a$ is a morphism $g: b \to b'$ such that $f = f' \circ g$. I don't see why this $g$ need be invertible when $a$ is minimal.

John Baez (Mar 13 2023 at 15:35):

Okay, now I see it.

Matteo Capucci (he/him) (Mar 13 2023 at 16:54):

Hugo Bacard said:

Matteo Capucci (he/him) said:

Here's a working definition:

A minimal object in $\cal C$ is an object $a : \cal C$ such that any morphism $b \to a$ is invertible

I might be wrong but is it equivalent to saying that the comma category $(\cal C \downarrow a)$ is a groupoid ?

Ha, nice! I was feeling something like that was in the air but I didn't pin it down yet.

Matteo Capucci (he/him) (Mar 13 2023 at 16:54):

Ah ok that's exactly what I expected actually -- that comma is $\mathcal C/a$