Let ${\mathcal{A}}$ be an abelian category with enough projectives. In the previous post, we described the definition of the derived ${\infty}$-category ${D^-(\mathcal{A})}$ of ${\mathcal{A}}$. As a simplicial category, this consisted of bounded-below complexes of projectives, and the space of morphisms between two complexes ${A_\bullet, B_\bullet}$ was obtained by taking the chain complex of maps ${\underline{Hom}(A_\bullet, B_\bullet)}$ between ${A_\bullet, B_\bullet}$ and turning that into a space (by truncation ${\tau_{\geq 0}}$ and the Dold-Kan correspondence).

Last time, we proved most of the following result:

Theorem 5 ${D^-(\mathcal{A})}$ is a stable ${\infty}$-category whose suspension functor is given by shifting by ${1}$. ${D^-(\mathcal{A})}$ has a ${t}$-structure whose heart is ${\mathcal{A}}$, and the homotopy category of ${D^-(\mathcal{A})}$ is the usual derived category.

Note for instance that this means that ${\mathcal{A}}$ sits as a full subcategory inside ${D^-(\mathcal{A})}$: that is, there is a full subcategory ${{D}^-(\mathcal{A})^{\heartsuit}}$ (the “heart”) of ${D^-(\mathcal{A})}$ (spanned by those complexes homologically concentrated in degree zero).

This heart has the property that the mapping spaces in ${D^-(\mathcal{A})^{\heartsuit}}$ are discrete, and the functor

$\displaystyle \pi_0: D^-(\mathcal{A}) \rightarrow \mathcal{A}$

restricts to an equivalence ${D^-(\mathcal{A})^{\heartsuit} \rightarrow \mathcal{A}}$; one can prove this by examining the chain complex of maps between two complexes homologically concentrated in degree zero. The inverse to this equivalence runs ${\mathcal{A} \rightarrow D^-(\mathcal{A})^{\heartsuit}}$, and it sends an element of ${\mathcal{A}}$ to a projective resolution. This is functorial in the ${\infty}$-categorical sense.

Most of the above theorem is exactly the same as the description of the ordinary derived category of ${\mathcal{A}}$ (i.e., the homotopy category of ${D^-(\mathcal{A})}$), The goal of this post is to describe what’s special to the ${\infty}$-categorical setting: that there is a universal property. I will start with the universal property for the subcategory ${D_{\geq 0}(\mathcal{A})}$.

Theorem 6 ${D_{\geq 0}(\mathcal{A})}$ is the ${\infty}$-category obtained from ${\mathcal{P} \subset \mathcal{A}}$ (the projective objects) by freely adding geometric realizations.

The purpose of this post is to sketch a proof of the above theorem, and to explain what it means. (more…)

The past few posts have been focused on a discussion of Lurie’s version of the Dold-Kan correspondence in stable $\infty$-categories. I’ve made these posts more detailed than usual: while I’ve been trying to treat such category theory as a black box on this blog, it should be interesting (at least for me) to see how the machines work beneath the surface, in some specific examples. In previous posts, I stated the result, and described an important lemma on the structure of simplicial objects in a stable $\infty$-category, which depended on the combinatorics of cubes.

The goal of this post is to (finally) prove the result, an equivalence of ${\infty}$-categories

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

valid for any stable ${\infty}$-category ${\mathcal{C}}$. As before, the intuition behind this version of the Dold-Kan correspondence is that a simplicial object determines a filtered object by taking successive geometric realizations of the ${n}$-truncations. The fact that one can go in reverse, and reconstruct the simplicial object from the geometric realizations of the truncations, is specific to the stable case. (more…)

Let ${\mathcal{C}}$ be a stable ${\infty}$-category. For us, this means that we have three important properties:

1. ${\mathcal{C} }$ admits finite limits and colimits.
2. ${\mathcal{C}}$ has a zero object: that is, the initial object is also final.
3. A square in ${\mathcal{C}}$ is a pull-back if and only if it is a push-out.

This is equivalent to the stability of ${\mathcal{C}}$. Actually, stability is usually defined using slightly weaker conditions, and then it takes a little work to show that these stronger ones are implied. We’ll just work with these. Stability can be thought of as a higher-categorical version of being triangulated; a general stable $\infty$-category has many of the properties (in a higher categorical sense) of the homotopy category of spectra, or the (classical) derived category.

Our goal is to show that in this case, we have an equivalence of ${\infty}$-categories

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

between simplicial objects in ${\mathcal{C}}$ and filtered (nonnegatively) objects in ${\mathcal{C}}$. The idea here is that the geometric realization of a simplicial object comes with a canonical filtration, given by geometric realizing the ${n}$-truncations for each ${n}$. This is going to give the associated filtered object. (We don’t know that the geometric realization exists, but the realizations of the truncations will.)

We will actually prove something stronger: for each ${n}$, there is an equivalence

$\displaystyle \mathrm{Fun}(\Delta^{op}_{\leq n}, \mathcal{C}) \simeq \mathrm{Fun}( [0, n], \mathcal{C}), \ \ \ \ \ (1)$

where ${[0, n] \subset \mathbb{Z}_{\geq 0}}$ is the subcategory of elements ${\leq n}$. In other words, ${n}$-truncated simplicial objects are the same as ${n}$-filtered objects of ${\mathcal{C}}$. (Note that, as a simplicial set, the nerve of ${[0, n]}$ is ${\Delta^n}$.) These equivalences will be compatible, and taking inverse limits will give the Dold-Kan correspondence. (more…)

Recall that we were in the middle of establishing a crucial equivalence of categories between simplicial abelian groups and chain complexes. Last time, we had defined the two functors: on the one hand, we had the normalized chain complex of a simplicial abelian group; on the other hand, we had defined a functor $\sigma$ that amalgamated a chain complex into a simplicial abelian group. We were in the middle of proving that the two functors were quasi-inverse.

With the same notation as before, we were trying to prove:

Proposition 3 (One half of Dold-Kan) For a simplicial abelian group ${A_\bullet}$, we have for each ${n}$, an isomorphism of abelian groups

$\displaystyle \bigoplus_{\phi: [n] \twoheadrightarrow [k]} NA_k \simeq A_n.$

Here the map is given by sending a summand ${NA_k}$ to ${A_n}$ via the pull-back by the term ${\phi: [n] \twoheadrightarrow [k]}$. Alternatively, the morphism of simplicial abelian groups

$\displaystyle \sigma (N A_*)_\bullet \rightarrow A_\bullet$

is an isomorphism.

Recall from last time that we were in the middle of proving the Dold-Kan correspondence, an important equivalence of categories between simplicial abelian groups and chain complexes. We defined three functors last time from simplicial abelian groups: the most obvious was the Moore complex, which just spliced all the components into one big chain complex with the differential the alternating sum of the face maps. But we noted that the functor one uses to construct this equivalence is ultimately either the normalized chain complex or the Moore complex modulo degeneracies.

Today, I’ll show that the two functors from simplicial abelian groups to chain complexes are in fact the same, through a decomposition that next time will let us construct the inverse functor. I’ll also construct the functor in the reverse direction. A minor word of warning: the argument in Goerss-Jardine (which seems to be the main source nowadays for this kind of material) has a small mistake! See their errata. This confused me for quite a while.

From chain complexes to simplicial groups

A priori, the normalized chain complex of a simplicial abelian group ${A_\bullet}$ looks a lot different from ${A_\bullet}$, which a priori has much more structure. Nonetheless, we are going to see that it is possible to recover ${A_\bullet}$ entirely from this chain complex. A key step in the proof of the Dold-Kan correspondence will be the establishment of the functorial decomposition for any simplicial abelian group ${A_\bullet}$

$\displaystyle \bigoplus_{\phi: [n] \twoheadrightarrow [k]} NA_k \simeq A_n. \ \ \ \ \ (1)$

Here the map from a factor ${NA_k}$ corresponding to some ${\phi: [n] \twoheadrightarrow [k]}$ to ${A_n}$ is given by pulling back by ${\phi}$. We will establish this below. (more…)

The next couple of posts will cover the Dold-Kan correspondence, which establishes an equivalence of categories between simplicial abelian groups and chain complexes. While this will not be strictly necessary for the introduction of the cotangent complex, it is a sufficiently important fact that it seems worth a digression.

As far as I can tell, the Dold-Kan correspondence is a fairly technical result, and I’m not sure I have any good intuition for why one should expect it to work. But at least one can say the following: given a simplicial abelian group (that is, a contravariant functor from the simplex category to the category of abelian groups), one can form a chain complex in a fairly easy manner: just take the $n$-simplices of the simplicial group as the degree $n$ part of the complex, and define the boundary using the alternating sum of the simplicial boundary maps (defined below); this is the classical computation that one does in introductory algebraic topology, of showing that the singular chain complex is indeed a complex.

So it’s natural that you would get a chain complex from a simplicial abelian group. Except, as it turns out, this is the wrong functor for the Dold-Kan correspondence; it is, however, close, being right up to (natural) homotopy.

The other bit of intuition that I’ve heard is the following. Given a topological space $X$, there is a means of obtaining the homology of $X$ as the homotopy groups of the infinite symmetric product; this is the so-called Dold-Thom theorem. (See this, for instance.) The Dold-Kan correspondence is in some sense the simplicial analog of this. The infinite symmetric product is much like the abelianization functor from simplicial sets to simplicial abelian groups (that applies the free abelian group pointwise). Now, it will come out of the Dold-Kan correspondence that the so-called “simplicial homotopy groups” of a simplicial abelian group are going to be the same thing as the homology of the associated chain complex. This is a rather loose analogy, and my understanding is that one cannot derive Dold-Thom from Dold-Kan. (more…)