The category of groupoids vs the category of sets

Question

Hopefully this question makes sense. As we know that Kan complexes are the "$\infty$-version" of groupoids for $\infty$-categories as groupoids for categories. On the other hand, the $\infty$-category of Kan complexes ($\mathrm{Anima}$) serves as the category of sets ($\mathrm{Set}$) for $\infty$-categories for many reasons. For example, we have Yoneda embedding by $\mathrm{Anima}$; there's a fully faithful functor $\mathrm{Set}\to \mathrm{Anima}$ etc.

My questions is: How should I understand the category of sets versus the category of groupoids as they both generalize to the category of Kan complexes? Also, can one use the category of groupoids to do Yoneda embedding?

Answer 1

The term $\infty$-groupoid is short, in older terminology, for weak infinity groupoid. There is an older version of (strict) infinity groupoid discussed by category theorists in the early 1970s. (These latter objects do not correspond to all homotopy types in a sense which would take too much space here.)

Some aspects of the history is worth sketching for this. Grothendieck in the long document Pursuing Stacks, (see the n-lab entry), proposed a higher dimensional analogue of structures known as 'stacks', better adapted for applications to problems of 'non-abelian cohomology'. These were in terms of some suggested model for (weak) infinity categories. 'Infinity stacks' were to be thought of as infinity categorical analogues of sheaves, and Grothendieck thought of those as generalising covering spaces (which are locally constant sheaves of sets). In a letter to Grothendieck (16 June 1983), I suggested that Kan complexes would satisfy his motivating conditions for infinity groupoids. (The term homotopy hypothesis is sometimes used for this, but Grothendieck's POV seems to have been that there would be many different models put forward for infinity groupoids and the test for a good model would be that of modelling all homotopy types.) In his reply to my letter, he was not that convinced by my viewpoint, as he was thinking in terms of a model which did not use simplices, but rather one involving `globes' (globular sets). (Such a model has since been worked out in a lot of detail.)

Some discussion of the history which tries not to assume much background in topology or algebraic geometry is given in the article here. The letters of Grothendieck should be published in a few months time including my letter from June 1983. The letters of Grothendieck to Larry Breen from 1975 are of particular note.

You asked about sets v. groupoids. Sets are just collections of things whilst grouopoids are collections of things with symmetries between them, therefore they can encode finer detail. There are 2-groupoids that model symmetries of symmetries in a certain sense, and the pattern can be continued.

As to the terminology, infinity groupoids are not Kan complexes if one approaches things from Grothendieck's direction. They are one model that works for his conditions. Kan complexes can be considered to be models of the idea of infinity groupoid, but they are not the only one, just a very convenient one. The idea of infinity groupoid grew out of algebraic topology, and algebraic geometry and not out of set theory. This is not a mishap nor an accident. In those subject areas, the structure is much more complex than with sets. Analogies with sets can be useful but so can ideas from algebraic and differential geometry, and especially from homotopy theory. Higher dimensional aspects of the structure of groupoids can be observed in the structure of infinity groupoids, but higher dimensional aspects of sets would be harder to pin down, so I am not convinced that $\infty$-sets would have been better term.

Answer 2

An important distinction in the "set case" is that there are two things you can refer to when talking about "the category of groupoids":

1. the 1-category of groupoids.
2. the 2-category of groupoids.

It is important to distinguishes these two: In (1) the correct notion of "sameness" is the isomorphism of groupoids. In (2) it is the equivalence of groupoids. And (2) is exactly the Dwyer-Kan localization of (1) and "weak equivalences" (= Fully faithful and essentially surjective functors).

In particular, because the notion of sameness" isn't the same in both, it is somehow incorrect to consider the two have the same objects. I'll call the objects of (1) "pregroupoids" ("strict groupoids" is also a common name).

Now both (1) and (2) have natural general to the $\infty$-categorical or animated setting:

For (1): It is a kind of algebraic structure, and we more or less know how to weaken these: An "animated pregroupoid" is the data of an "Anima of objects" X, with "for each $x,y \in X$ an anima of morphisms $Hom(x,y)$" (I mean by that a functor $X \times X \to An$) equipped with a composition operation and an inverse operation that satisfies all the expected higher associativity and coherence conditions.

There are various way to make this definition formal (The simplest one being probably using the theory of Segal spaces) but at the end of day you can show (with more or less work depending on the definition you use) that there is an equivalence of category:

$$ \text{Animated pregroupoids} \simeq S \subset An^{\to} $$

Where $S$ is the full subcategory of things that are surjective on $\pi_0$. An arrow $f: X \to Y$ surjective on $\pi_0$ corresponds to the animated pregroupoids that has $X$ for anima of objects and for each $x,y\in X$ as $Hom(x,y)$ being the objects of path between $f(x)$ and $f(y)$ in $Y$.

So this is the analogue of (1) in the $\infty$-categorical setting.

Now for (2) : it corresponds to localizing the category at the appropriate notion of "weak equivalences".

If you think about it for a second, you'll see that the equivalence are exactly the square $(X \to Y) \to (X' \to Y')$ where the map $Y \to Y'$ is a weak equivalence. And so, contrary to the set theoretical case, when you localize the $\infty$-category of animated pregroupoids this way you can describe the localization very explicitly because the localization corresponds to a reflection (each $X \to Y$ goes to $Y \to Y$), and the localization is just the category of Anima. (so the $\infty$-categorical analogue of $(2)$ is just the category of Anima).

To some extent the fact that in the "animated" setting the "bicategory of (animated) groupoids" and the "category of set (=anima)" end-up being the same is a big reason (if not the main) why on many respect the theory of $(\infty,1)$-category behave better than ordinary category theory.