What is curvature, really?

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 be a manifold, and let be a vector bundle. Suppose given a connection on . This determines, and is equivalent to, the data of parallel transport along each (smooth) curve . In other words, for each such , one gets an isomorphism of vector spaces

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 , we get a map

that sends a loop at to the induced automorphism of given by parallel transport along it. (Here we’ll want to take to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.)

 

Acyclic spaces I

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 -category. Dror didn’t have this language available, but his results fit neatly into it.

Let be the -category of pointed spaces. We have a functor

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 is . We can look at the subcategory consisting of spaces sent by to zero (that is, to a contractible complex).

 

Definition 1 Spaces in are called acyclic spaces.

 

The subcategory is closed under colimits (as is colimit-preserving). It is in fact a very good candidate for a homotopy theory: that is, it is a presentable -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.

 

The universal property of the derived infinity-category

Let be an abelian category with enough projectives. In the previous post, we described the definition of the derived -category  of . As a simplicial category, this consisted of bounded-below complexes of projectives, and the space of morphisms between two complexes was obtained by taking the chain complex of maps between and turning that into a space (by truncation and the Dold-Kan correspondence).

Last time, we proved most of the following result:

 

Theorem 5  is a stable -category whose suspension functor is given by shifting by . has a -structure whose heart is , and the homotopy category of is the usual derived category.

 

Note for instance that this means that sits as a full subcategory inside : that is, there is a full subcategory (the “heart”) of (spanned by those complexes homologically concentrated in degree zero).

This heart has the property that the mapping spaces in are discrete, and the functor

restricts to an equivalence ; 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 , and it sends an element of to a projective resolution. This is functorial in the -categorical sense.

Most of the above theorem is exactly the same as the description of the ordinary derived category of (i.e., the homotopy category of ), The goal of this post is to describe what’s special to the -categorical setting: that there is a universal property. I will start with the universal property for the subcategory .

 

Theorem 6  is the -category obtained from (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.

 

Understanding the derived infinity-category

The next thing I’d like to do on this blog is to understand the derived -category of an abelian category.

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

For instance, , and given a triangulated category and a functor taking exact sequences in to triangles in , we might want there to be an extended functor

where is the ordinary (1-categorical) bounded derived category of . We might expect this by the following rough intuition: given an object of we can represent it as obtained from objects in by taking a finite number of cofibers and shifts. As such, we should take the image of to be the appropriate combination of cofibers and shifts in of the images of . 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 -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: is obtained from the category of projective objects in by freely adjoining geometric realizations. In other words:

Theorem 1 (Lurie) Let be an abelian category with enough projectives, which form a subcategory . Then has the following property. Let be any -category with geometric realizations; then there is an equivalence

between the -categories of functors and geometric realization-preserving functors .

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 higher-categorical Dold-Kan correspondence II

The past few posts have been focused on a discussion of Lurie’s version of the Dold-Kan correspondence in stable -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 -category, which depended on the combinatorics of cubes.

The goal of this post is to (finally) prove the result, an equivalence of -categories

valid for any stable -category . 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 -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.

 

The cube lemma in a stable infinity-category

Let be a stable -category. In the previous post, we needed to consider cubical diagrams

These diagrams come with an initial object and a terminal object: in fact, they are the cones on smaller diagrams. For instance, is the nerve of all subsets of , which is the cone on the nerve of all nonempty subsets of , and also the cone on the nerve of all proper subsets of . So it makes sense to talk about whether is a limit diagram, or whether is a colimit diagram.

The main result is:

Proposition 11 (Cube lemma) If is stable, then is a limit diagram if and only if it is a colimit diagram.

 

When , this is automatic: any diagram is a limit diagram if and only if it is an equivalence, and ditto for colimit diagrams. When , 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.

 

The higher-categorical Dold-Kan correspondence

Let be a stable -category. For us, this means that we have three important properties:

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

This is equivalent to the stability of . 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 -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 -categories

between simplicial objects in and filtered (nonnegatively) objects in . The idea here is that the geometric realization of a simplicial object comes with a canonical filtration, given by geometric realizing the -truncations for each . 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 , there is an equivalence

where is the subcategory of elements . In other words, -truncated simplicial objects are the same as -filtered objects of . (Note that, as a simplicial set, the nerve of is .) These equivalences will be compatible, and taking inverse limits will give the Dold-Kan correspondence.

 

More examples of cofinality

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

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

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, looks like . This is not contractible: if we take 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, is contractible. We’ll see that this is true in general for any , but the 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 -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.

 

Simplicial objects in infinity-categories

Let be an -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 -category of spectra. A simplicial object of is a functor

that is, it is a morphism of simplicial sets from the nerve of the opposite of the simplex category to . 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 be a simplicial set, thus defining a homotopy type and thus an object of the -category of spaces. Alternatively, can be regarded as a simplicial object in sets, so a simplicial object in (discrete) spaces. In other words, has two incarnations:

1. .
2. .

The connection is that is the geometric realization (in the -category of spaces) of the simplicial object . More generally, whenever one has a bisimplicial set , defining an object of , then the geometric realization of in is the diagonal simplicial set . These are model categorical observations: one chooses a presentation for (e.g., the usual Kan model structure on simplicial sets), and then uses the fact that -categorical colimits in 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 -category “really” means, though, so let’s do some more examples.

 

Segal’s approach to delooping

(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 . 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 : the construction of the space . It is known that any topological group is (weakly) homotopy equivalent , 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 -idea. Given a topological group , we can construct using a standard simplicial construction. If 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 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 , the simplicial set has the following properties.

1. is a point.
2. The map induced by the inclusions (sending and 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 is such a simplicial set, then we take as the underlying set of the monoid, and the map comes from the boundary map ; the identity element comes from the map . 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.”