Let’s do some more examples of cofinality. In the previous post, I erroneously claimed that the map

$\displaystyle \Delta^{op}_{inj \leq n} \rightarrow \Delta^{op}_{\leq n}$

was cofinal: that is, taking a colimit of an ${n}$-truncated simplicial object in an ${\infty}$-category was the same as taking the colimit of the associated ${n}$-truncated semisimplicial object. (The claim has since been deleted.) This is false, even when ${n = 1}$. In fact, the map of categories

$\displaystyle \Delta^{op}_{inj \leq 1} \rightarrow \Delta^{op}_{\leq 1}$

is not even a weak homotopy equivalence. (While this is not obvious, one of the statements of “Theorem A” is that a cofinal map is automatically a homotopy equivalence.)

In fact, ${\Delta^{op}_{inj \leq 1}}$ looks like ${\bullet \rightrightarrows \bullet}$. This is not contractible: if we take ${\pi_1}$ of the nerve, that’s the same as taking the category itself and inverting all the morphisms. So that gives us a free groupoid on two morphisms. However, ${\Delta^{op}_{\leq 1}}$ is contractible. We’ll see that this is true in general for any ${\Delta^{op}_{\leq n}}$, but the $\Delta^{op}_{inj, \leq n}$ only become “asymptotically” contractible.

The purpose of this post is to work through a few examples of Theorem A, discussed in the previous post. This will show that the colimit of an $n$-truncated simplicial object can in fact be recovered from the semisimplicial restriction, but in a somewhat more subtle way than one might expect. We will need this lemma in the discussion of the Dold-Kan correspondence. (more…)

Let ${\mathcal{C}}$ be an ${\infty}$-category, in the sense of Joyal and Lurie: in other words, a quasicategory or weak Kan complex. For instance, for the purposes of Hopkins-Miller, we’re going to be interested in the ${\infty}$-category of spectra. A simplicial object of ${\mathcal{C}}$ is a functor

$\displaystyle F: N(\Delta^{op}) \rightarrow \mathcal{C} ,$

that is, it is a morphism of simplicial sets from the nerve of the opposite ${\Delta^{op}}$ of the simplex category to ${\mathcal{C}}$. A geometric realization of such a simplicial object is a colimit. A simplicial object is like a reflexive coequalizer (in fact, the 1-skeleton is precisely a reflexive coequalizer diagram) but with extra “higher” data in bigger degrees. Since reflexive coequalizers are a useful tool in ordinary category theory (for instance, in flat descent), we should expect geometric realizations to be useful in higher category theory. That’s what this post is about.

A simple example of a geometric realization is as follows: let ${X_\bullet}$ be a simplicial set, thus defining a homotopy type and thus an object of the ${\infty}$-category ${\mathcal{S}}$ of spaces. Alternatively, ${X_\bullet}$ can be regarded as a simplicial object in sets, so a simplicial object in (discrete) spaces. In other words, ${X_\bullet}$ has two incarnations:

1. ${X_\bullet \in \mathcal{S}}$.
2. ${X_\bullet \in \mathrm{Fun}(\Delta^{op}, \mathcal{S})}$.

The connection is that ${X_\bullet}$ is the geometric realization (in the ${\infty}$-category of spaces) of the simplicial object ${X_\bullet}$. More generally, whenever one has a bisimplicial set ${Y_{\bullet, \bullet}}$, defining an object of ${\mathrm{Fun}(\Delta^{op}, \mathcal{S})}$, then the geometric realization of ${Y_{\bullet, \bullet}}$ in ${\mathcal{S}}$ is the diagonal simplicial set ${n \mapsto Y_{n, n}}$. These are model categorical observations: one chooses a presentation for ${\mathcal{S}}$ (e.g., the usual Kan model structure on simplicial sets), and then uses the fact that ${\infty}$-categorical colimits in ${\mathcal{S}}$ are the same as model categorical colimits in simplicial sets. Now, it is a general fact from model category theory that the homotopy colimit of a bisimplicial set is the diagonal.

So we can think of all homotopy types as being built up as geometric realizations of discrete ones. I’ve been trying to understand what a simplicial object in an ${\infty}$-category “really” means, though, so let’s do some more examples. (more…)

(This is the second post devoted to unpacking some of the ideas in Segal’s paper “Categories and cohomology theories.” The first is here.)

Earlier, I described an observation (due to Beck) that loop spaces could be characterized as algebras over the monad ${\Omega \Sigma}$. At least, any loop space was necessarily an algebra over that monad, and conversely any algebra over that monad was homotopy equivalent to a loop space. There is an alternative and compelling idea of Segal which gives a condition somewhat easier to check.

As far as I understand, most of the different approaches to delooping a space consist of imitating the classical construction for a topological group ${G}$: the construction of the space ${BG}$. It is known that any topological group ${G}$ is (weakly) homotopy equivalent ${\Omega BG}$, and conversely (though perhaps it is not as well known) that any loop space is homotopy equivalent to a topological group. (This can be proved using the simplicial construction of Kan.) Given a space (which may not be a topological group), the idea is that delooping machinery will assume given just enough structure to build something analogous to the classifying space, and then build that. This is, for instance, how the construction of Beck ran.

Here’s Segal’s idea; it is quite similar to the ${\Gamma}$-idea. Given a topological group ${G}$, we can construct ${BG}$ using a standard simplicial construction. If ${G}$ is only a group object in the homotopy category, we can’t run this construction. Segal decides just to assume that one has given the data of a simplicial object that behaves like ${BG }$ should and runs with that.

The starting point is that one can encode the structure of a monoid in a simplicial set. Given a monoid ${G}$, the simplicial set ${BG}$ has the following properties.

1. ${(BG)_0}$ is a point.
2. The map ${(BG)_n \rightarrow \prod_{i=1}^n (BG)_1}$ induced by the ${n}$ inclusions ${[1] \rightarrow [n]}$ (sending ${0}$ and ${1}$ to consecutive elements) is an isomorphism.

In fact, if we have any simplicial set with the above properties, it determines a unique monoid. This is proved in a similar way. If ${X_\bullet}$ is such a simplicial set, then we take ${X_1}$ as the underlying set of the monoid, and the map ${X_1 \times X_1 \rightarrow X}$ comes from the boundary map ${X_2 \rightarrow X_1}$; the identity element comes from the map ${X_0 = \ast \rightarrow X_1}$. So monoids can be described as simplicial sets satisfying certain properties (just as commutative monoids can).

As before, we can weaken this by replacing “isomorphism” by “homotopy equivalence.” (more…)

There are a lot of forms of the Brown representability theorem, which all basically assert that a functor on a suitable homotopy category which plays well with arbitrary coproducts and satisfies a weak condition on push-outs, is representable.

The form proved by Brown was the following. Let ${\mathbf{hCW}}$ be the homotopy category of pointed CW complexes

Theorem 1 (Brown, c. 1950) Let ${F: \mathbf{hCW} \rightarrow \mathbf{Sets}}$ be a contravariant functor such that ${F}$ sends coproducts to products. Suppose that if ${(X_1, X_2, A) \subset X}$ is a proper triad—i.e., that ${(X_1, A)}$ and ${(X_2, A)}$ were CW pairs with ${X_1 \cap X_2 = A, X_1 \cup X_2 = X}$—then the map

$\displaystyle F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)$

is surjective. Then ${F}$ is representable. (more…)

I’ve been a bit busy and consequently behind on this blog, but I thought I would advertise some notes I wrote, covering chapter 1 of “Higher Topos Theory.” I’m giving a talk on them at a seminar tomorrow, and I wrote them mostly to prepare, but I tried to be detailed.