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Stream: learning: questions

Topic: How to characterize a groupoid by its equipment in Cat

Asad Saeeduddin (Apr 15 2021 at 17:03):

Here is my amateurish definition of a groupoid: a groupoid is a category G for which there is a functor inv : G -> G^op that is the identity on objects, such that for any morphism p : G(A, B), inv(p) . p = idA and p . inv(p) = idB

Now I don't want to say things like "functor that is the identity on objects" and so on, but I'm not sure how to avoid it. Is there a way I can characterize a groupoid as an object in Cat that is equipped with some additional 1,2-cells and equations?

John Baez (Apr 15 2021 at 17:45):

This is a sort of famous problem, that becomes more serious for dagger categories: a generalization of groupoids where you have a functor $\dagger$ that's the identity on objects, for which $\dagger \circ \dagger$ is the identity. People worry about whether saying "the identity on objects" is evil:

• Are dagger categories truly evil, MathOverflow.

• Principle of equivalence - for dagger-categories, nLab.

So, I think reading the whole discussions there would be good.

John Baez (Apr 15 2021 at 17:49):

By the way, if you really want to avoid saying "functor that is the identity on objects", how do you plan to define the identity functor?

Asad Saeeduddin (Apr 15 2021 at 18:09):

@John Baez i'd say it's the neutral element of functor composition (i.e. it is the identity endomorphism of each object in Cat)

Asad Saeeduddin (Apr 15 2021 at 18:09):

i hope that somehow avoids talking about the objects of the category itself and the "action" of the functor on them. but maybe that isn't really accomplishing anything beyond talking about objects directly, i'm not sure

Mike Shulman (Apr 15 2021 at 19:36):

John Baez said:

By the way, if you really want to avoid saying "functor that is the identity on objects", how do you plan to define the identity functor?

That's a different question -- when we define the identity functor we can define it to act by the identity function on objects. It doesn't require referring to an equality relation between two given objects, which is what one needs to say, as a property of a hypothesized functor, that it acts as the identity on objects.

Mike Shulman (Apr 15 2021 at 19:42):

Asad Saeeduddin said:

Is there a way I can characterize a groupoid as an object in Cat that is equipped with some additional 1,2-cells and equations?

A groupoid is an object $G\in \rm Cat$ together with, for every pair of 1-cells $x,y:1\to G$ and 2-cell $\alpha : x\Rightarrow y$, a 2-cell $\beta:y\Rightarrow x$ such that $\alpha\beta = 1$ and $\beta\alpha=1$.

If you don't like giving a special role to the terminal category 1 (e.g. if you want to generalize to other 2-categories), you can assert the same condition for 1-cells $x,y:X\to G$ for an arbitrary object $X$. This works just as well in Cat, and in any 2-category gives the correct notion of groupoidal object.

If you don't like quantifying over all objects $X$ of the category, you can say that $G$ is a groupoid if the map $G \to G^{\mathbf{2}}$ is an equivalence, where $\mathbf{2}$ is the interval category. This works in any 2-category that admits powers by $\mathbf{2}$, and gives the same notion of groupoidal object.

David Michael Roberts (Apr 16 2021 at 01:26):

Could you get the same from demanding that $G$ satisfies the universal property for the power by $\mathbf{2}$?

Mike Shulman (Apr 16 2021 at 03:37):

Yeah. But that gets you back to quantifying over all objects of the category, whereas if there already are powers then you can consider them a finitary operation. (-:

Asad Saeeduddin (Apr 19 2021 at 17:05):

@Mike Shulman beautiful, thanks very much