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: community: general

Topic: Group object in an arbitrary category

David Michael Roberts (Oct 18 2020 at 23:13):

Motivated by a slightly ill-posed question on MathOverflow, I was wondering if one could make sense of a group object in an arbitrary category? The best I can do is a category with a terminal object. Then the relevant finite products must just be demanded to exist for the object in question. This feels a bit like the recent post by @John Baez on possibly empty groups: these gizmos could certainly be defined internal to an arbitrary category, as they only need binary products, and these are in some sense a "local" requirement (we only need certain binary products to exist).

An alternative, and to some extent, the cop-out answer, is to say that an internal group in an arbitrary category is an internal transitive groupoid $G$. Transitivity can be defined by asking that $G_1 \to G_0\times G_0$ is an extremal epi, or perhaps a regular epi. (Aside: contrary to apparently popular thought, internal groupoids make sense in an arbitrary category; one again just needs to demand that certain pullbacks exist.) However, one doesn't get the right category of groups this way ... :-/

Another alternative is to ask that one has a possibly empty group, but that the carrier object is not initial. This is rather weak, since the object could be subterminal. In the slice category $Set/X$, such a thing could be a bundle $p\colon G\to XX$ of possibly empty groups over $X$, such that there is at least one $x\in X$ such that $p^{-1}(x)$ is an honest group. In particular, the inclusion $\{x\} \to X$ is a bundle of possibly empty groups, where only one of these is not empty—and it's trivial!

Any thoughts?

Nathanael Arkor (Oct 19 2020 at 00:20):

I'm not sure it's fair to say that you can interpret a group object in just a category with a terminal object, because, as you point out, you need certain finite products to exist: why should a terminal object be part of the assumption, but not the other finite products? From this point of view, it seems like you're saying that a group object can be interpreted in any category with a group object (after having unwound the definition), though this is tautological. (You can always obfuscate the requirement that there be a terminal object, e.g. "A group object in a category C is a model of the Lawvere theory of groups in C.".)

Nathanael Arkor (Oct 19 2020 at 00:32):

Alternatively, you can eliminate the terminal object condition by asking for constant morphisms from every object in the category, i.e. unwrapping the definition of a morphism 1 → G when there is no terminal object.

Reid Barton (Oct 19 2020 at 00:33):

You can always formally define a group object in C as an object X of C together with a group structure on the presheaf represented by X. I'm not sure whether such a thing can exist without the products of finitely many copies of X existing in C.

David Michael Roberts (Oct 19 2020 at 01:54):

It makes perfect sense to define a group object in $Mfld/M$, despite this category not having all binary products: these are just bundles of Lie groups, which people do care about. One might just ask that the map to $M$ is a submersion, so that the product exists, for instance, or more narrowly, that the bundle is locally trivial.

David Michael Roberts (Oct 19 2020 at 01:56):

I did wonder about constant morphisms, but it can be tricky to get these to behave as they ought, in very odd examples. I like the idea of having a constant morphism from every object in the category, though. There was a MathOverflow discussion about constant maps (and on the HoTT mailing list, and at the nForum....and probably here, too) that gave some options how one can sensibly think about this both via presheaves, and in terms of the category itself.

Morgan Rogers (he/him) (Oct 19 2020 at 08:26):

David Michael Roberts said:

However, one doesn't get the right category of groups this way ... :-/

It sounds like you're hoping for the usual category of groups in familiar situations? If you're going to impose that restriction, why shouldn't the consequence be that there simply aren't any group objects in a category without a terminal object? The sketch for groups expresses the minimal limit and morphism structure required to find a group object even in a category without all finite products.

Eric M Downes (Mar 24 2024 at 02:41):

This thread is 4 years old, and maybe what I have to say is obvious / useless to everyone else, but what the hell.

I think a group object can be expressed in a (multi?)category $\mathsf{C}$ without products. Example: This is sufficient to express a semigroup object $S\in obj.\mathsf{C}$:
Look ma, no products!
So the equations for $obj.\mathsf{C}\ni S;~\mu,ж:S\to S$ are:
$ж\circ ж= id_S$
$\mu\circ\mu=ж\circ\mu\circ\mu\circ ж$

Admittedly, my proof-sketch for sufficiency uses string diagrams and $S=\mathsf{X\otimes X\otimes X}$.
stringified semigroup law
Can you establish a functor to the desired object from a symmetric monoidal category? ...
That might be way worse than the problem it purports to solve! :)

I don't actually know what slice category $\mathsf{Mfld/M}$ is, but essentially we smuggle a copy of $C_2$ inside the illicit semigroup where most Algebra Enforcement Detectives haven't been trained to look for products. If that is not too repugnant, I believe similar stringification might work for the inverse and identity.

Eric M Downes (Mar 24 2024 at 03:50):

Ok, the coherence requirements I can come up with for inverses seem to require a notion of $S$ being a "weak product object" (my made up term, which caveat, may not be weak at all); it has three $\pi_i:S\to S$:
$\pi_i\circ\pi_j=\begin{cases}\eta & i\neq j\\ \pi_i & i=j\end{cases}$

and then you have something like the following for each $\zeta_i$, with transpositions in $S_3$ relating all the $\zeta_i$ to one another. inverse dia and stringies
There are other coherence rules which I haven't written down.

I think the way I'm writing the $\pi_i$'s gives them the universal property we're hoping to avoid, because every incoming morphism that mutates one of the three "parts" alone, can factor through the appropriuate $\pi_i$. Essentially I am wondering if the only such objects that have this internal product structure are just obfuscated products, or if there is any genuine content here. ... I'd imagine its something like in group theory

$M,N\unlhd G~~\&~~M\cap N=\set1\implies G\cong G/M\times G/N$

that is, if you have sufficiently independent congruences within an object, that is equivalent to the object having an internal product, even if your ambient category lacks products. Maybe that is too hopeful... if a category lacks products surely a generic object would also be subdirectly irreducible? I guess its back to the same "if the products exist for this object" thing.

So in conclusion

• I believe semigroup objects can be expressed without products! Possibly not group objects.
• Q: Can an object $S\in\mathsf{C}$ be a $\mathsf{D}$-object, where $\mathsf{D}$ does have a product, without $S$'s factors also being objects in $\mathsf{C}$?