### Cocycle Category

#### Posted by Urs Schreiber

Here is another guest post by Bruce Bartlett.

Luckily, Bruce is still at Fields in Toronto, attending the Thematic Program on Geometric Applications of Homotopy Theory.

Here he reports on something very interesting that is intimately related to our discussion of anafunctors.

Hi guys,

I’m no genius in homotopy theory but I think I stumbled across a really cool fact today which I’d like to bounce off you, perhaps you’d like to put it on the $n$-café. I’m so excited about it, I almost want to keep it secret… except that if true, it’s probably too small an observation to warrant a paper.

As you know, Rick Jardine is giving lectures on Simplicial Presheaves.

For $n$-café purposes, some of these notes can be found in the preprints
section of his homepage. I have
also taken notes and perhaps I can scan them in. But the important point
is that the entire lecture today was based on his *Cocycle categories*
preprint which can be found at that link.

What he does is he considers a model category $M$. Given two objects $X$ and $Y$ in $M$, you define a cocycle from $X$ to $Y$ as a pair of maps

$\array{ Z &\to& Y \\ \downarrow \\ X } \,,$ where the map to $X$ is a weak equivalence. A morphism between cocycles is just an obvious map making the obvious diagram commute. Then $H(X,Y)$ is called the cocycle category.

Of course, us $n$-category theorists recognize this immediately - he is simply building the 2-category $\mathrm{NiceSpan}(M)$ of ‘nice’ spans in $M$, whose objects are the same as $M$, whose morphisms are cocycles (spans where the projection map is a weak equivalence) and whose 2-morphisms are 2-morphisms between spans! (We know this because Urs taught us this from one of his recent posts at the n-cafe). Rick Jardine never mentions the word 2-category, though.

[*The definition of the ordinary bicategory of spans is recalled here, aspects of the 2-category of anafunctors are discussed here. For more details check out M. Makkai’s paper and Toby Bartels’ thesis. -urs*]

Here’s the great fact. He proves that for decent model categories $M$ (when $M$ is a right proper closed model structure in which weak equivalences are preserved by finite products), the canonical map $\phi : \pi_0 H(X,Y) \to \mathrm{Ho}(M) (X,Y)$ is a bijection! Here $\pi_0 H(X,Y)$ is just the iso classes of morphisms, and $\mathrm{Ho}(M)$ is the homotopy category of $M$. The map $\phi$ sends $(f,g)$ to $f^{-1} g$ (you know what I mean).

As $n$-café regulars, we get excited because we immediately see two things:

(1) He has shown that the homotopy category $\mathrm{Ho}(M)$ is nothing but $\pi_1$ of the 2-category $\mathrm{NiceSpan}(M)$! (I know that there is already a ‘$\pi_1$ of a 2-category’ interpretation of $\mathrm{Ho}(M)$, but doesn’t that only work for cofibrant fibrant objects?)

(2) It is obvious that in the ordinary model structure on $\mathrm{Cat}$, the
cocycles $X \to Y$ are precisely the anafunctors! In other words,
**morphisms in the homotopy category $\mathrm{Ho}(\mathrm{Cat})$ are nothing but isomorphism
classes of anafunctors**!!!

Finally, if you read his preprint above on *cocycle categories*, you’ll
see that the entire motivating example is nothing but the cocycle
associated to a principal $G$-bundle… in other words *it’s nothing but
the stuff you guys needed to do for transport*. This is why
‘anafunctors’ etc comes out so naturally.

I’m really excited about this… but I’m going to sleep now. Hopefully you guys will bring me down gently if I’ve gotten confused here, or if its all old hat.

Yours, Bruce

## Re: Cocycle Category

He hasn’t quite finished building this until he defines composition of cocycles. This is straightforward, of course, using pullbacks just as in the ordinary composition of spans. While you need all pullbacks to define composition of arbitrary spans, here you only need pullbacks involving weak equivalences, but you also need that the pullback of any weak equivalence be a weak equivalence. (I forget if this condition is part of the definition of model category, but I won’t be surprised if it is.)

I don’t find any of this surprising, although it’s certainly good that people are thinking about it. My concern with model categories (which I’d heard of before going to Toronto this month but which always seemed too complicated to learn, to the point that I’d never even looked carefully at the definition) as a tool for understanding higher categories (not that they are only that) is that they are more complicated than necessary for that purpose. Much can be done with only a good notion of weak equivalence. I don’t know if there any axiomatisations of the concept of category-with-weak-equivalences.