# Tag Info

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 ...
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### 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 ...
• 59.8k

### 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 ...
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### 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 ...
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• 31.3k

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

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 ...
• 3,153
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 $$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$ ...
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### 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 $n\geq 1$...
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### 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 ...
• 29.2k
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### 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 ...
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