### category theory

I’ve been reading an interesting paper of Adams, Haeberly, Jackowski, and May on the Atiyah-Segal completion theorem. One of the surprising features of this paper is the heavy use of pro-abelian groups to deal with the inconvenient fact that inverse limits are generally not exact in abelian groups. I’d like to blog about the proof in this paper, but first I’d like to go through some of the background on pro-objects. In this post, I’ll describe the entirely dual picture of ${\mathrm{Ind}}$-objects, which is (at least for me) easier to understand.

1. Definition

Let ${\mathcal{A}}$ be a small abelian category. Then there is an imbedding

$\displaystyle \mathcal{A} \hookrightarrow \mathrm{Ind}(\mathcal{A}),$

of ${\mathcal{A}}$ into the larger category of ind-objects of ${\mathcal{A}}$. One benefit of doing this is that ${\mathrm{Ind}(\mathcal{A})}$ is a larger abelian category containing ${\mathcal{A}}$, in which there are enough injectives.

I always found the traditional definition of these confusing, so let me describe another definition (which happens to generalize nicely to the ${\infty}$-categorical case, and which is where I learned it from).

Let ${\mathcal{C}}$ be any category. Then we know that the category ${P(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{op}, \mathbf{Sets})}$ is the “free cocompletion” of ${\mathcal{C}}$: that is, given any cocomplete category ${\mathcal{D}}$, we have an equivalence

$\displaystyle \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \simeq \mathrm{Fun}^{L}( P(\mathcal{C}), \mathcal{D})$

between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and colimit-preserving functors ${P(\mathcal{C}) \rightarrow \mathcal{D}}$. The ${\mathrm{Ind}}$-category is defined to have an analogous universal property, except that one just takes filtered colimits. (more…)

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 next thing I’d like to do on this blog is to understand the derived ${\infty}$-category of an abelian category.

Given an abelian category ${\mathcal{A}}$ with enough projectives, this is a stable ${\infty}$-category ${D^-(\mathcal{A})}$ with a special universal property. This universal property is specific to the ${\infty}$-categorical case: in the ordinary derived category of an abelian category (which is the homotopy category of ${D^-(\mathcal{A})}$), forming cofibers is not quite the natural process it is in ${D^-(\mathcal{A})}$ (in which it is a type of colimit), and one cannot expect the same results.

For instance, ${\mathcal{A}}$, and given a triangulated category ${\mathcal{T}}$ and a functor ${\mathcal{A} \rightarrow \mathcal{T}}$ taking exact sequences in ${\mathcal{A}}$ to triangles in ${\mathcal{T}}$, we might want there to be an extended functor

$\displaystyle D_{ord}^b(\mathcal{A}) \rightarrow \mathcal{T},$

where ${D_{ord}^b(\mathcal{A})}$ is the ordinary (1-categorical) bounded derived category of ${\mathcal{A}}$. We might expect this by the following rough intuition: given an object ${X}$ of ${D^b(\mathcal{A})}$ we can represent it as obtained from objects ${A_1, \dots, A_n}$ in ${\mathcal{A}}$ by taking a finite number of cofibers and shifts. As such, we should take the image of ${X}$ to be the appropriate combination of cofibers and shifts in ${\mathcal{T}}$ of the images of ${A_1, \dots, A_n}$. Unfortunately, this does not determine a functor because cofibers are not functorial or unique up to unique isomorphism at the level of a trinagulated category.

The derived ${\infty}$-category, though, has a universal property which, among other things, makes very apparent the existence of derived functors, and which makes it very easy to map out of it. One formulation of it is specific to the nonnegative case: ${D_{\geq 0}(\mathcal{A})}$ is obtained from the category of projective objects in ${\mathcal{A}}$ by freely adjoining geometric realizations. In other words:

Theorem 1 (Lurie) Let ${\mathcal{A}}$ be an abelian category with enough projectives, which form a subcategory ${\mathcal{P}}$. Then ${D_{\geq 0}(\mathcal{A})}$ has the following property. Let ${\mathcal{C}}$ be any ${\infty}$-category with geometric realizations; then there is an equivalence

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

between the ${\infty}$-categories of functors ${\mathcal{P} \rightarrow \mathcal{C}}$ and geometric realization-preserving functors ${D_{\geq 0}(\mathcal{A}) \rightarrow \mathcal{C}}$.

This is a somewhat strange (and non-abelian) universal property at first sight (though, for what it’s worth, there is another more natural one to be discussed later). I’d like to spend the next couple of posts understanding why this is such a natural universal property (and, for one thing, why projective objects make an appearance); the answer is that it is an expression of the Dold-Kan correspondence. First, we’ll need to spend some time on the actual definition of this category.

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. In the previous post, we needed to consider cubical diagrams

$\displaystyle f: (\Delta^1)^{n+1} \rightarrow \mathcal{C}.$

These diagrams come with an initial object and a terminal object: in fact, they are the cones on smaller diagrams. For instance, ${(\Delta^1)^{n+1}}$ is the nerve of all subsets of ${[n]}$, which is the cone on the nerve of all nonempty subsets of ${[n]}$, and also the cone on the nerve of all proper subsets of ${[n]}$. So it makes sense to talk about whether ${f}$ is a limit diagram, or whether ${f}$ is a colimit diagram.

The main result is:

Proposition 11 (Cube lemma) If ${\mathcal{C}}$ is stable, then ${f: (\Delta^1)^{n+1} \rightarrow \mathcal{C}}$ is a limit diagram if and only if it is a colimit diagram.

When ${n = 0}$, this is automatic: any diagram ${\Delta^1 \rightarrow \mathcal{C}}$ is a limit diagram if and only if it is an equivalence, and ditto for colimit diagrams. When ${n = 1}$, this is particular to the stable case: a square is a push-out if and only if it is a pull-back. We took this as more or less axiomatic, though it can be deduced from much weaker axioms, as in “Higher Algebra.”

The purpose of this post is to work through the proof of the “cube lemma.” This is more or less a piece of an attempt to work through Lurie’s version of the Dold-Kan correspondence. I’ve been doing it in a fair bit of detail for my own benefit — this means that the posts are a little more detailed than usual. In any event, the present post should stand alone from the others. (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…)

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…)