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.

1. The Dold-Kan correspondence

Let ${\mathcal{A} }$ be an ordinary abelian category. Then one has the Dold-Kan correspondence (see e.g. this post)

$\displaystyle \mathrm{Fun}(\Delta^{op}, \mathcal{A}) \simeq \mathrm{Ch}_{\geq 0}(\mathcal{A})$

between simplicial objects in ${\mathcal{A}}$ and chain complexes in ${\mathcal{A}}$ of nonnegative degree. (Chain complexes are graded homologically: the differential has degree ${-1}$.) I’ve always found this result rather mysterious: the relevant functors in the Dold-Kan equivalence are complex and not very intuitive. There are various ${\infty}$-categorical versions of the Dold-Kan correspondence, though, which I’ve been trying to learn about.

In “Higher Algebra,” Lurie defines a higher-categorical version of the derived category. Given an abelian category ${\mathcal{A}}$ with enough projectives, the (non-classical) derived category ${D^-(\mathcal{A})}$ of ${\mathcal{A}}$ is an ${\infty}$-category whose objects are bounded-below complexes of projectives. There is a chain complex of morphisms between any two such complexes, which gives a space by taking the Dold-Kan correspondence. In this way, one can extract an ${\infty}$-category. As usual, there is a truncated version ${D_{\geq 0}(A)}$ consisting of complexes (homologically) concentrated in nonnegative degrees.

Theorem 1 (Lurie) Let ${\mathcal{A}}$ be an abelian category with enough projectives. Then ${D_{\geq 0}(\mathcal{A})}$ is freely generated by the projectives of ${\mathcal{A}}$ under geometric realizations.In other words, if ${\mathcal{P} \subset \mathcal{A}}$ is the (ordinary) subcategory of projectives, and ${\mathcal{C}}$ is any ${\infty}$-category admitting geometric realizations, then there is an equivalence

$\displaystyle \mathrm{Fun}'(D_{\geq 0}(\mathcal{A}), \mathcal{C}) \simeq \mathrm{Fun}(\mathcal{P}, \mathcal{C}),$

where ${\mathrm{Fun}'}$ denotes the subcategory of functors preserving geometric realizations.

So we should think of any nonnegatively graded chain complex as determining a geometric realization of objects in ${\mathcal{P}}$, and the nonnegatively graded derived category can alternatively be described as some sort of free cocompletion (with respect to geometric realizations) of the projective objects.

This isn’t actually what Lurie calls the “Dold-Kan correspondence” in his book: that label is applied to the following result.

Theorem 2 (Lurie) Given a stable ${\infty}$-category ${\mathcal{C}}$, there is a canonical equivalence

$\displaystyle \mathrm{Fun}( \mathbb{Z}_{\geq 0}, \mathcal{C}) \simeq \mathrm{Fun}(\Delta^{op}, \mathcal{C})$

between the ${\infty}$-categories of simplicial objects of ${\mathcal{C}}$ and filtered objects of ${\mathcal{C}}$.

For various reasons, a filtered object in a stable ${\infty}$-category (e.g., spectra or the derived category) is the natural replacement for a “chain complex.” In fact, one can show that a filtered object determines a “chain complex” in the (triangulated) homotopy category. The idea here is that a simplicial object determines a filtered object by taking successive skeleta (i.e., by attaching only cells in degrees $\leq n$). In the stable setting, one can go in reverse, and this is a special property of the combinatorics of $\Delta$.

2. Theorem A

To start with, let’s derive some results on simplicial objects themselves. For instance, we’d like to have a result such as the following, to enhance computability and intuition.

Proposition 3 Let ${F: \Delta^{op} \rightarrow \mathcal{C}}$ be a simplicial object in an ${\infty}$-category ${\mathcal{C}}$. Then the geometric realization of ${F}$ is equivalent to the colimit of the semisimplicial object ${F|_{\Delta^{op}_{inj}}}$, where one restricts to injective morphisms in ${\Delta}$.

This is a “cofinality” result: one can take a colimit either over the whole simplex category or over the “semisimplex” category, where we don’t include the degeneracies. To prove something like this, there’s a general set of tools in “Higher Topos Theory.” Given a map of ${\infty}$-categories ${f: \mathcal{C} \rightarrow \mathcal{D}}$, say that it is cofinal if taking a colimit over ${\mathcal{D}}$ is equivalent to taking a colimit over ${\mathcal{C}}$. One has the following useful criterion:

Theorem 4 (Theorem A) A map ${f: \mathcal{C}\rightarrow \mathcal{D}}$ between ${\infty}$-categories is cofinal if and only if the ${\infty}$-categories ${\mathcal{C} \times_{\mathcal{D}} \mathcal{D}_{d/}}$ are weakly contractible for each ${d \in \mathcal{D}}$.

Why is this a natural result to expect? One reason is the following: if ${f: \mathcal{C} \rightarrow \mathcal{D}}$ is cofinal, then surely, for any ${\mathcal{D}}$-indexed diagram in spaces, one might as well compute the colimit over ${\mathcal{D}}$ or over ${\mathcal{C}}$. Let’s fix an object ${d \in \mathcal{D}}$, and consider the corepresentable functor

$\displaystyle h_d: \mathcal{D} \rightarrow \mathcal{S}, \quad x \mapsto \hom_{\mathcal{D}}(d, x).$

Then if we have cofinality, we must surely have

$\displaystyle \varinjlim_{\mathcal{D}} h_d \simeq \varinjlim_{\mathcal{C}} h_d \circ F.$

The amazing story here is that these fairly abstract colimits admit a concrete geometric model, and this gives us half of the theorem. Namely, there’s this “Grothendieck construction” which identifies (i.e., under a Quillen equivalence of model categories) the following two objects:

1. Functors ${\mathcal{D} \rightarrow \mathcal{S}}$ into the category of spaces.
2. “Left fibrations” ${X \rightarrow \mathcal{D}}$. A left fibration is the ${\infty}$-categorical analog of a category cofibered in groupoids.

The classical Grothendieck construction states that, for an ordinary category ${C}$, to give a functor ${C \rightarrow \mathrm{Gpd}}$ equates to giving a category cofibered in groupoids over ${C}$. Lurie’s version states that the same is true in ${\infty}$-category land, except that “groupoids” are replaced by spaces.

Moreover, the point of relevance here is that the Grothendieck construction gives an efficient means for writing down the colimit of a functor ${G: \mathcal{D} \rightarrow \mathcal{S}}$: namely, you have to take the associated left fibration ${\mathcal{D}' \rightarrow \mathcal{D}}$, and the colimit is just ${\mathcal{D}'}$ (considered as a homotopy type). In the case when one works with a representable functor ${h_d: \mathcal{D} \rightarrow \mathcal{S}}$, the associated left fibration is the overcategory

$\displaystyle \mathcal{D}_{d/} \rightarrow \mathcal{D}.$

The upshot of all this is that, in order for ${f: \mathcal{C} \rightarrow \mathcal{D}}$ to be cofinal, the map

$\displaystyle \mathcal{C} \times_{\mathcal{D}} \mathcal{D}_{d/} \rightarrow \mathcal{D}_{d/}$

has to be a weak homotopy equivalence. The left side corresponds (under the Grothendieck construction) to the colimit of ${h_d \circ F}$, while the right-hand-side corresponds to the colimit of ${h_d}$. But ${\mathcal{D}_{d/}}$ has an initial object and is thus weakly contractible.

This at least shows that cofinality implies the conclusion of Quillen’s Theorem A. In practice, one wants to go in reverse, but it’s at least not hard to believe now that if ${\mathcal{C} \times_{\mathcal{D}} \mathcal{D}_{d/} \rightarrow \mathcal{D}_{d/} }$ is always a weak equivalence, then ${f: \mathcal{C} \rightarrow \mathcal{D}}$ is cofinal: we’ve seen that the weak equivalence condition means that ${f}$ is cofinal “on representable functors,” and those generate the category of all functors (at least into spaces). As it happens, making that precise is done later and not earlier in HTT, so maybe we should just take this as motivation.

3. Semisimplicial objects

Let’s now prove (fully!) the result advertised earlier: given a simplicial object ${F: \Delta^{op} \rightarrow \mathcal{C}}$ in the ${\infty}$-category ${\mathcal{C}}$, then the colimit of ${F}$ is the same as the colimit of the associated semisimplicial object. In other words, we have to show that if ${\Delta^{op}_{inj}}$ is the semisimplex category (no degeneracies), then

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

is cofinal.

How can we do this? Theorem A tells us that we need to show that for any ${n \in \mathbb{Z}_{\geq 0}}$, the category ${\Delta_{inj}^{op} \times_{\Delta^{op}} \Delta_{[n]/}^{op}}$ is (weakly) contractible. This category can be described as follows:

1. An object is a pair ${([m], f)}$ where ${m \in \mathbb{Z}_{\geq 0}}$ and ${f: [m] \rightarrow [n]}$ is a morphism.
2. A morphism between ${( [m], f)}$ and ${( [m'], g)}$ is an injective map ${[m'] \rightarrow [m]}$ making the relevant diagram of maps to ${[n]}$ commute.

To show that this category, which I’ll call ${\mathcal{D}}$, is weakly contractible, we will exhibit an endofunctor ${F: \mathcal{D} \rightarrow \mathcal{D}}$ together with natural transformations (homotopies!)

$\displaystyle 1_{\mathcal{D}} \leftarrow F \rightarrow c$

where ${c}$ is a constant functor. This will mean that the identity map of ${\mathcal{D}}$ is nullhomotopic.

In fact, fix an inclusion ${[0] \hookrightarrow [n]}$, and take ${F}$ to be the functor

$\displaystyle F: [m] \mapsto [m] \sqcup [0],$

where ${[m] \sqcup [0]}$ maps to ${[n]}$ via the fixed inclusion ${[0] \hookrightarrow [n]}$. There is a natural transformation ${1_{\mathcal{D}} \rightarrow F}$, coming from the natural inclusion ${[m] \hookrightarrow [m] \sqcup [0]}$. Moreover, there is a natural transformation ${F \rightarrow c}$, corresponding to the natural inclusion

$\displaystyle [0] \hookrightarrow [m] \sqcup [0].$

Together, these furnish the required natural transformations, and enable us to apply Theorem A to conclude the cofinality result. We can actually say a little more. (Edit: this refers to a mistaken assertion which is now deleted.)