### category theory

This post is part of a series on the Sullivan conjecture in algebraic topology. The Sullivan conjecture is a topological result, which remarkably reduces — as H. Miller showed —  to a purely algebraic computation in the category of unstable modules (and eventually algebras) over the Steenrod algebra, and in particular an injectivity assertion. This is a rather formidable category, but work of Kuhn enables one to identify a quotient category of it with the category of “generic representations” of the general linear group, which can be studied using different (and often easier) means. Kuhn’s work provides an approach to proving much of the algebraic background that goes into the Sullivan conjecture. In this post, I’ll describe one of the important ingredients.

The Gabriel-Popsecu theorem is a structure theorem for Grothendieck abelian categories, a version of which will be useful in understanding the structure of the category of unstable modules over the Steenrod algebra. The purpose of this post is to discuss this result and its many-object version due to Kuhn, from the paper “Generic Representations of the Finite General Linear Groups and the Steenrod Algebra: I.” Although the proof consists mostly of a series of diagram chases, there are some subtleties that I found rather difficult to grasp, and I thought it would be worthwhile to go through it in detail here.

1. Grothendieck abelian categories

Let ${\mathcal{A}}$ be an abelian category. Then ${\mathcal{A}}$ is Grothendieck abelian if

• ${\mathcal{A}}$ has a generator: that is, there is an object ${X \in \mathcal{A} }$ such that every object in ${\mathcal{A}}$ can be built up from colimits starting with ${X}$. (More precisely, the smallest subcategory of ${\mathcal{A}}$, closed under colimits, that contains ${X}$ is ${\mathcal{A}}$ itself)
• Filtered colimits in ${\mathcal{A}}$ exist and are exact.

Many categories occurring in “nature” (e.g., categories of modules over a ring of sheaves on a site) are Grothendieck, and it is thus useful to have general results about them. The goal of this post is to describe a useful structure theorem for Grothendieck abelian categories, which will show that they are the quotients of categories of modules by Serre subcategories. (more…)

Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.

1. Curvature of a connection

Let ${M }$ be a manifold, and let ${V \rightarrow M}$ be a vector bundle. Suppose given a connection ${\nabla}$ on ${V}$. This determines, and is equivalent to, the data of parallel transport along each (smooth) curve ${\gamma: [0, 1] \rightarrow M}$. In other words, for each such ${\gamma}$, one gets an isomorphism of vector spaces

$\displaystyle T_{\gamma}: V_{\gamma(0)} \simeq V_{\gamma(1)}$

with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.

In particular, if we fix a point ${p \in M}$, we get a map

$\displaystyle \Omega_p M \rightarrow \mathrm{GL}( V_p)$

that sends a loop at ${p}$ to the induced automorphism of ${V_p}$ given by parallel transport along it. (Here we’ll want to take ${\Omega_p M}$ to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.) (more…)

I recently read E. Dror’s paper “Acyclic spaces,” which studies the category of spaces with vanishing homology groups. It turns out that this category has a fair bit of structure; in particular, it has a theory resembling the theory of Postnikov systems. In this post and the next, I’d like to explain how the results in Dror’s paper show that the decomposition is really a special case of the notion of a Postnikov system, valid in a general ${\infty}$-category. Dror didn’t have this language available, but his results fit neatly into it.

Let ${\mathcal{S}}$ be the ${\infty}$-category of pointed spaces. We have a functor

$\displaystyle \widetilde{C}_*: \mathcal{S} \rightarrow D( \mathrm{Ab})$

into the derived category of abelian groups, which sends a pointed space into the reduced chain complex. This functor preserves colimits, and it is in fact uniquely determined by this condition and the fact that ${\widetilde{C}_*(S^0)}$ is ${\mathbb{Z}[0]}$. We can look at the subcategory ${\mathcal{AC} \subset \mathcal{S}}$ consisting of spaces sent by ${\widetilde{C}_*}$ to zero (that is, to a contractible complex).

Definition 1 Spaces in ${\mathcal{AC}}$ are called acyclic spaces.

The subcategory ${\mathcal{AC} \subset \mathcal{S}}$ is closed under colimits (as ${\widetilde{C}_*}$ is colimit-preserving). It is in fact a very good candidate for a homotopy theory: that is, it is a presentable ${\infty}$-category. In other words, it is a homotopy theory that one might expect to describe by a sufficiently nice model category. I am not familiar with the details, but I believe that the process of right Bousfield localization (with respect to the class of acyclic spaces), can be used to construct such a model category. (more…)

I’ve been reading Wall’s “Finiteness conditions for CW complexes.” This paper gives necessary and sufficient conditions for a space to be homotopy equivalent to a finite cell complex. Alternatively, it gives an obstruction in ${K}$-theory for when a retract (in the homotopy category) of a finite cell complex has the homotopy type of a finite cell complex. I’d like to describe this result, and try to motivate why the existence of such an obstruction is a natural thing to expect by a simpler analogy with algebra.

There is a fruitful analogy between spaces and chain complexes. Let ${R}$ be a ring, and consider the derived category ${D(R)}$ of chain complexes of ${R}$-modules. There are various interesting subcategories of ${D(R)}$:

1. The finitely presented derived category ${D_{fp}(R)}$; this is the smallest triangulated (or stable) full subcategory of ${D(R)}$ containing ${R}$ and closed under cofiber sequences. In other words, ${D_{fp}(R)}$ consists of complexes which are quasi-isomorphic to finite complexes of finitely generated free modules.
2. The perfect derived category ${D_{pf}(R)}$: this is the category of objects ${X \in D(R)}$ such that ${\hom(X, \cdot)}$ commutes with direct sums (i.e., the compact objects). It turns out that so-called perfect complexes are those that can be represented as finite complexes of finitely generated projectives.

One should think of the finitely presented objects as analogous to the finite cell complexes in topology, and the perfect objects as analogous to the retracts of finite cell complexes. (To push the analogy: the finite cell complexes are the smallest subcategory of the ${\infty}$-category of spaces containing ${\ast}$ and closed under finite colimits. The retracts of finite cell complexes are the compact objects in this ${\infty}$-category.) (more…)

Let ${A}$ be a regular local (noetherian) ring with maximal ideal ${\mathfrak{m}}$ and residue field ${k}$. The purpose of this post is to construct an equivalence (in fact, a duality)

$\displaystyle \mathbb{D}: \mathrm{Mod}_{\mathrm{sm}}(A) \simeq \mathrm{Mod}_{\mathrm{sm}}(A)^{op}$

between the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$-modules (i.e., finitely generated modules annihilated by a power of ${\mathfrak{m}}$) and its opposite. Such an anti-equivalence holds in fact for any noetherian local ring ${A}$, but in this post we will mostly stick to the regular case. In the next post, we’ll use this duality to give a description of the local cohomology groups of a noetherian local ring. Most of this material can be found in the first couple of sections of SGA 2 or in Hartshorne’s Local Cohomology.

1. Duality in the derived category

Let ${A}$ be any commutative ring, and let ${\mathrm{D}_{\mathrm{perf}}(A)}$ be the perfect derived category of ${A}$. This is the derived category (or preferably, derived ${\infty}$-category) of perfect complexes of ${A}$-modules: that is, complexes containing a finite number of projectives. ${\mathrm{D}_{\mathrm{perf}}(A)}$ is the smallest stable subcategory of the derived category containing the complex ${A}$ in degree zero, and closed under retracts. It can also be characterized abstractly: ${\mathrm{D}_{\mathrm{perf}}(A)}$ consists of the compact objects in the derived category of ${A}$. That is, a complex ${X}$ is quasi-isomorphic to something in ${\mathrm{D}_{\mathrm{perf}}(A)}$ if and only if the functor

$\displaystyle \hom(X, \cdot) : \mathrm{D}(A) \rightarrow \mathbf{Spaces}$

commutes with homotopy colimits. (“Chain complexes” could replace “spaces.”) (more…)

Let ${X}$ be a quasi-compact, separated scheme. Then a criterion of Serre asserts that ${X}$ is affine if and only if

$\displaystyle H^i(X, \mathcal{F}) =0 , \quad i > 0,$

for all quasi-coherent sheaves ${\mathcal{F}}$ on ${X}$: that is, affine schemes are characterized by the vanishing of the higher cohomology of all quasi-coherent sheaves. The purpose of this post is to explain an interpretation of Serre’s theorem (or rather, the “if” direction) in terms of category theory. Namely, the idea is that if ${X}$ satisfies the cohomological vanishing condition, then the functor

$\displaystyle \Gamma: \mathrm{QCoh}(X) \rightarrow \mathrm{Mod}(\Gamma(X,\mathcal{O}_X)),$

from the category of quasi-coherent sheaves on ${X}$ to the category of modules over ${\Gamma(X, \mathcal{O}_X)}$, turns out to be a symmetric monoidal equivalence for formal reasons. A version of Tannakian formalism now shows that ${X}$ is itself isomorphic to ${\mathrm{Spec} \Gamma(X, \mathcal{O}_X)}$: that is, the category ${\mathrm{QCoh}(X)}$ together with its symmetric monoidal structure recovers ${X}$.

Edit: I was sure this material was well-known folklore, but didn’t have a reference when I posted this. Now I do; see Knutson’s Algebraic Spaces.

Second edit (11/19): I just realized that the argument below leaves out an important piece: it doesn’t show that the structure sheaf is a generator! I’ll try to fix this soon.

The Atiyah-Segal completion theorem calculates the ${K}$-theory of the classifying space ${BG}$ of a compact Lie group ${G}$. Namely, given such a ${G}$, we know that there is a universal principal ${G}$-bundle ${EG \rightarrow BG}$, with the property that ${EG}$ is contractible. Given a ${G}$-representation ${V}$, we can form the vector bundle

$\displaystyle EG \times_G V \rightarrow BG$

via the “mixing” construction. In this way, we get a functor

$\displaystyle \mathrm{Rep}(G) \rightarrow \mathrm{Vect}(BG),$

and thus a homomorphism from the (complex) representation ring${R(G)}$ to the ${K}$-theory of ${BG}$,

$\displaystyle R(G) \rightarrow K^0(BG).$

This is not an isomorphism; one expects the cohomology of an infinite complex (at least if certain ${\lim^1}$ terms vanish) to have a natural structure of a complete topological group. Modulo this, however, it turns out that:

Theorem (Atiyah-Segal) The natural map ${R(G) \rightarrow K^0(BG)}$ induces an isomorphism from the ${I}$-adic completion ${R(G)_{I}^{\wedge} \simeq K^0(BG)}$, where ${I}$ is the augmentation ideal in ${R(G)}$. Moreover, ${K^1(BG) =0 }$.

The purpose of this post is to describe a proof of the Atiyah-Segal completion theorem, due to Adams, Haeberly, Jackowski, and May. This proof uses heavily the language of pro-objects, which was discussed in the previous post (or rather, the dual notion of ind-objects was discussed). Remarkably, their approach uses this formalism to eliminate almost all the actual computations, by reducing to a special case. (more…)

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.

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