## April 29, 2010

### Understanding the Homotopy Coherent Nerve

#### Posted by Mike Shulman

(guest post by Emily Riehl)

Simplicial categories, more properly called simplicially enriched categories, provide a model for $\left(\infty ,1\right)$-categories which is related to quasi-categories by means of an adjunction $ℂ:$sSet $↔$sSet-Cat$:ℕ$. This adjunction induces a Quillen equivalence between André Joyal’s model structure on simplicial sets and Julie Bergner’s model structure on simplicial categories. Consequently, this adjunction serves as a primary means of translating between the two worlds, figuring prominently, for example, in the straightening construction, which is the $\left(\infty ,1\right)$-analog of the classical Grothendieck construction of a contravariant Cat-valued pseudofunctor from a categorical fibration.

The main disadvantage to this perspective is that the functor $ℂ$, although defined explicitly as a pointwise left Kan extension, is not easily understood – or rather, was not easily understood until a recent paper by Dugger and Spivak. Their “necklace characterization,” detailed below, makes it easy to compute the simplicial categories associated to simplicial sets as well as prove a number of results, at least one of which is rather surprising.

We’ll begin by reviewing the definition of the homotopy coherent nerve $ℕ$, which we present, following Dugger and Spivak, by means of a comonad resolution, a perspective we find more intuitive than the usual one. We’ll then turn our attention to the left adjoint $ℂ$ and explain the details of the necklace characterization. At the conclusion, we’ll give a number of applications, most of which can be found in more detail here.

Posted at 3:03 AM UTC | Permalink | Followups (4)

### Confessions of a Higher Category Theorist

#### Posted by Mike Shulman

This is a (long-delayed) continuation of the series begun by Urs and Tom a while ago, about various approaches to higher category theory and how we feel about its current and future directions. Specifically, I want to talk about the new “$\left(\infty ,-\right)$-ized” category theory that’s rapidly spreading through mathematics, based on models like quasicategories and complete Segal spaces. As Urs has eulogized at length, it’s really exciting that we now seem to have notions of higher category where one can really develop a theory and start to actually do and apply higher category theory.

On the other hand, I think there are still questions and open directions in higher category theory that aren’t addressed or solved by these models, and thus there are reasons to continue working on alternate models. In particular, I’m not yet ready to declare that quasicategories are The Way to talk about $\left(\infty ,1\right)$-categories; they seem to be quite good for now, but there are things about them that one may still want to improve on. Now I don’t mean to imply that anyone has said that they are The Way (although if anyone out there does feel that way, let’s talk about it!). Rather, what I’d like to do is really have a discussion about what’s good and what’s not as good about these models, and what directions still remain for foundational progress in higher category theory.

Posted at 2:53 AM UTC | Permalink | Followups (27)

## April 26, 2010

### This Week’s Finds in Mathematical Physics (Week 296)

#### Posted by John Baez

In week296 of This Week’s Finds, you can get a free book on how Felix Klein used the icosahedron to solve the quintic equation. And then we’ll try to construct a compact dagger-category where the morphisms are electrical circuits made of resistors!

Posted at 6:54 PM UTC | Permalink | Followups (32)

## April 21, 2010

### On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

#### Posted by Simon Willerton

Back in my recent post on intrinsic volumes I said that I was just finishing a paper on the magnitude of certain Riemannian manifolds. It is now essentially finished.

I will put it on the arxiv next week: comments and suggestions are welcome.

Posted at 7:33 AM UTC | Permalink | Followups (32)

## April 17, 2010

### This Week’s Finds in Mathematical Physics (Week 295)

#### Posted by John Baez

In week295 of This Week’s Finds, learn about the principle of least power, Poincaré duality for electrical circuits — and a curious generalization of Hamiltonian mechanics that involves entropy as well as energy. Also, check out the eruption of Eyjafjallajökull!

Posted at 1:36 AM UTC | Permalink | Followups (32)

## April 15, 2010

### Paris in the Spring

#### Posted by David Corfield

I’m just back from a couple of days of philosophy of mathematics workshops in Paris, which turns out to be one of the liveliest places on the planet for the subject. Had I realised, I might have extended my trip by a day to take in the first screening of Edward Frenkel’s Rites of Love and Math.

I was invited to contribute to the second day on the theme of the ‘complexity of proof’ by Andrew Arana. I’m rather doubtful that one can give anything resembling a measure of the complexity of a proof, although I did hear some interesting ideas from Paul-André Melliès and Alessandra Carbone who are looking to assign invariants to formal proofs, group theoretic ones for the latter.

Posted at 12:29 PM UTC | Permalink | Followups (29)

## April 8, 2010

### Symmetric Monoidal Bicategories and (n×k)-categories

#### Posted by Mike Shulman

I just put a note on the arXiv about constructing symmetric monoidal bicategories. The definition of symmetric monoidal bicategory is somewhat imposing, but in many cases all the structure can be “lifted” for free from a symmetric monoidal double category, which is a much easier structure. For instance, the bicategories of spans, cobordisms, and profunctors all inherit their monoidal structures in this way. Here’s the link:

The tricky part is that in a monoidal double category, the coherence isomorphisms such as $X\otimes \left(Y\otimes Z\right)\cong \left(X\otimes Y\right)\otimes Z$ are vertical isomorphisms, but these are the morphisms that get discarded when we pass to the horizontal bicategory. Thus, we need to be able to “lift” these isomorphisms to horizontal equivalences, in such a way that we can ensure coherence is preserved. The structure we need is essentially that of a proarrow equipment (regarded as a double category), although in this context, with attention focused on the “bicategory of proarrows” rather than the 2-category of arrows, it seems more appropriate to call it a framed bicategory or a fibrant double category. Richard Garner and Nick Gurski proved essentially the same result in section 5 of this paper, using “locally-double bicategories;” the machinery I used also makes braiding and symmetry easy to deal with.

Posted at 6:01 AM UTC | Permalink | Followups (26)

## April 6, 2010

### New Paper on the Hecke Bicategory

#### Posted by Alexander Hoffnung

Hi everyone!

I have just finished a draft of an expository style paper called The Hecke Bicategory on groupoidified Hecke algebras and permutation representations. This paper is a companion to the latest papers in the Higher Dimensional Algebra series – HDA 7: Groupoidification and HDA 8: The Hecke Bicategory (in progress).

I would be happy to hear comments and corrections (no matter how small).

The paper is available on my website and the arXiv.

Posted at 7:44 PM UTC | Permalink | Followups (17)