Tag Info

65

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 once spelled out explicitly. Moreover being younger than Voevosky I have never been really exposed to the idea that categories were sets of higher dimension (but ...

39

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 early work on higher categories was "wrong"), I want to add that while Voevodsky is of course correct from a certain point of view, there is another valid point ...

27

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 answers: namely, that "being strict" is a property of $\infty$-groupoids, rather than a structure put on an $\infty$-groupoid. It is reasonable to suppose we ...

17

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 obstruction as mentioned in the comment), but rather that the construction that attach a convolution $C^*$-algebra to a groupoid is not 'injective' at all. This is ...

16

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 $\prod_{i\geq 1}\frac{1}{(1-x^{5i-4})(1-x^{5i-1})}$ by the first Rogers-Ramanujan identity. You can also interpret the second Rogers-Ramanujan identity as giving a ...

14

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 was an influence; he had visited Chicago for a year some time earlier. The use of groupoids in algebra was common knowledge in the 1940s, see the 1943 book on ...

14

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 Eilenberg MacLane spaces. $$X \simeq \prod_{n \in \mathbb N} K(\pi_i X, n).$$ This is a consequence of the Dold-Kan correspondence, and the fact that every ...

12


8

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 conical pseudolimits and filtered pseudolimits commute in $\mathbf{Cat}$; whereas the latter proves the analagous result for finite weighted bilimits and ...

7

Actions of a Lie groupoid are defined on p. 34 of K.C.H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids" LMS Lecture Notes Series no 213, 2005. Note that in general for a groupoid action of $G$ on sets it is often convenient to follow C. Ehresmann and to insist on having a function $f: E \to Ob(G)$ so that $g \in G(x,y)$ maps the fibre of ...

7

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 For a singular foliation see http://users.uoa.gr/~iandroul/AS-holgpd-final.pdf

7

I don't think it's a good idea to mix the slogans "topological spaces are locales" and "topological spaces are $\infty$-groupoids." I think the former slogan encapsulates what we ended up defining as topological spaces while the latter slogan encapsulates what we should've defined as topological spaces, at least if we're algebraic topologists (recall that $\... 7 Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between$\mathbb{R}$-coefficients and$\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story. The cohomology of the continuous cochains is ... 7 I am a novice too, but I think that an equivalent perspective (which is the first I met) is far more enlightening than the Kan condition. Maybe it is the right moment to check if I'm right: things can't be so simple! :) Start with your favourite small category$\bf C$. Denote its set of objects as${\bf C}_0$. Now consider the set of arrows$X\to Y$; ... 7 The fundamental groupoid does not preserve homotopy limits. The simplest examples can be obtained from the path-space sequence$\Omega X\to \operatorname{pt}\to X$which shows that$\Omega X$is the homotopy limit of$\operatorname{pt}\to X\leftarrow \operatorname{pt}$. If$\pi_1$preserves homotopy limits, then the fundamental groupoid of$\Omega X\$ should ...

Only top voted, non community-wiki answers of a minimum length are eligible