67
votes

Accepted

### Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?

First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial ...

40
votes

### Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?

The other answers are quite good and nothing is wrong with them, but lest a wrong impression be given (e.g. by the implicit suggestion that the idea of "categories as sets in the next dimension" in ...

28
votes

### How do you define the strict infinity groupoids in Homotopy Type Theory?

I realize I'm dropping in long after all the excitement is over. However, I'd like to point out an implicit assumption which is (seemingly) inherent in the original question as well as some of the ...

17
votes

Accepted

### Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?

The main obstruction to this kind of duality is not so much that not every $C^*$-algebra is a convolution algebra (though, at least if we don't use twisted convolution algebra, there are known ...

16
votes

Accepted

### What is the groupoid cardinality of the category of vector spaces over a finite field?

Upon substituting $x=\frac{1}{q}$ we obtain
$$\sum_{n\geq 0}\frac{1}{|\mathrm{GL}_n(\mathbb F_q)|}=\sum_{n\geq 0}\frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}$$
and this evaluates to the product $\prod_{i\...

15
votes

### Brandt's definition of groupoids (1926)

The influence of Brandt's groupoid definition on the definition of category by Eilenberg and Mac Lane has been discussed on the category discussion list.
Bill Cockcroft told me in 1964-70 that there ...

15
votes

### Which $\infty$-groupoids correspond to simplicial abelian groups?

Basically, you're asking which (weak) homotopy types can be modelled by simplicial sets which carry the structure of a simplicial abelian group.
Every simplicial abelian group splits as a product of ...

12
votes

### Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?

Even if this answer can be seen as an expansion of the last two lines of Simon's answer, it does not really come in the same spirit.
From the point of view of enriched category theory, posets are ...

12
votes

### Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and
hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-...

11
votes

### Has any attempt been made to classify finite groupoids?

Everything that's been written so far about the classification of finite groupoids reducing to the classification of finite groups is true but, I think, misleading. In order to actually produce a list ...

10
votes

Accepted

### Toposes (topoi) as classifying toposes of groupoids

Perhaps these slides will be helpful. I'll try to explain what happens in your special case.
Let $M$ be a monoid and let $\mathcal{B} M$ be the topos of right $M$-sets. The points of $\mathcal{B} M$ ...

10
votes

Accepted

### On fundamental groupoid of fundamental groupoid

The discussion at the linked nlab page points to a few different accounts of topologies on the fundamental groupoid, but I'm not feeling energetic enough to track them all down.
So let me just ...

9
votes

### Coverings of a space and coverings of a groupoid

As you will see this is a very rich area, so my answer can do no more than give a sketch. The basic idea is that a good category of coverings, Cov(X), say, has certain good categorical properties and ...

9
votes

### Which $\infty$-groupoids correspond to simplicial abelian groups?

$\newcommand{\Z}{\mathbf{Z}} \newcommand{\Mod}{\mathrm{Mod}}$Note that if $\Mod^{\geq 0}_\Z$ denotes the category of connective $\mathrm{H}\Z$-module spectra, then by the Dold-Kan correspondence and ...

8
votes

Accepted

### Groupoid cardinality and Egyptian fraction representations of 1

First a notational issue: you shouldn't write $G$ for the one-object groupoid corresponding to $G$. A much better name for this groupoid is $BG$, or $\text{pt} / G$.
A natural way to write down a ...

8
votes

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

The fundamental groupoid does not preserve homotopy limits. The simplest examples can be obtained from the path-space sequence $\Omega X\to \operatorname{pt}\to X$ which shows that $\Omega X$ is the ...

8
votes

### Classification of weak 3-groups

Yes, but it's more complicated. First let me describe the special case where $\pi_1$ is trivial. If $G$ is such a 3-group, then $X = BG$ is a pointed connected simply connected space. It fits into a ...

8
votes

Accepted

### What are Lie groupoids intuitively?

Here is an expansion of my comment, by request. The 2-category of Lie groupoids $\mathrm{LieGpd}$ admits the category of manifolds $\mathrm{Mfld}$ as a full sub-2-category (i.e. $\mathrm{Mfld} \to \...

8
votes

I cannot say what exactly Voevodsky meant but here is a wild guess.
Disclaimer in what follows I use heavily type theoretic notation, so you have trouble understanding feel free to ask in the ...

8
votes

Accepted

### Is the 2-сategory of groupoids locally presentable?

This is true.
Since the $2$-category of groupoids is equivalent to the $2$-category of $1$-truncated spaces, the statement follows immediately from 5.5.1.8 and 5.5.6.21 in Lurie's Higher Topos ...

8
votes

Accepted

### Filtered 2-colimits commute with finite 2-limits

Two relevant papers are:
Dupont's Interchange of filtered 2-colimits and finite 2-limits.
Canevali's 2-filtered bicolimits and finite weighted bilimits commute in Cat.
The former proves that finite ...

7
votes

Accepted

### How to construct a free 2-group on a groupoid?

let me offer you an alternative. The free 2-group on a groupoid is well defined up to equivalence (not isomorphism), hence I'll offer you a strict model (which is known to exist by abstract reasons, ...

7
votes

### What's a groupoid? What's a good example of a groupoid?

The holonomy groupoid of a foliation is another example of a useful groupoid
it is described here:
http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01036-5/S0273-0979-04-01036-5.pdf
...

Community wiki

7
votes

Accepted

### moduli stack of double covers of $\mathbb{P}^1$ with one marked point

It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but ...

7
votes

### What are Lie groupoids intuitively?

In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a symultaneous generalization of group actions
$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$
...

7
votes

Accepted

### Space with semi-locally simply connected open subsets

$X$ need not be locally simply connected. Consider the following construction:
Let $X_0=S^1$ and $X_n=CS^1$ for all $n\geq 1$ where $CS^1$ is the cone over the circle. Let $Y_0=X_0$, and for $n\geq 1$...

7
votes

Accepted

### Homotopy of functors

I'm surprised this has been up for days with nobody telling Atticus the obvious. Let $I$ be the unit interval category with two objects, $0$ and $1$, and one non-identity arrow $0 \to 1$. A natural ...

7
votes

Accepted

### Representation of fundamental groupoid as $2$-sheaf

First of all, note that you haven't used the fact that $\Pi_1(-)$ was the terminal $2$-cosheaf, just that it was a ($2$-)cosheaf.
Then, as I pointed out in the comments, there's a question of whether ...

7
votes

### Elementary theory of the category of groupoids?

Robert Harper and Dan Licata studied the topic under the name 2-dimensional type theory, see their Canonicity for 2-Dimensional Type Theory and possibly these
slides. As they are computer scientists ...

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