What is a deformation of a category?

Question

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.

What is a deformation of a (linear, dg, A-infinity) category? Is it a "bundle of categories" over a scheme? How can you make such a notion rigorous? Maybe via stacks? Suppose we take some nice scheme $X$ and we consider $D^b\text{Coh}(X)$; if we deform $X$, then do we also get a corresponding deformation of the derived category? What does this corresponding deformation "look like"? Are we deforming the morphisms? The objects? Both?

Similarly, what about in the situation where we have a category of modules over an algebra $A$? If we deform the algebra, then do we also get a corresponding deformation of the category? Again, what does it "look like"?

And finally, in either of the above cases, are there deformations of the respective categories that don't correspond to deformations of $X$ or of $A$ respectively? I expect the answer to be "yes"; then my next question is: Are there any nice examples of such deformations that can still be described in an explicit or geometric way?

I am most of all interested in concrete examples, and less interested in general theory.

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Edit 1: I probably should have mentioned this when I first posted this (almost 3 months ago now!), but somehow I forgot. Kontsevich has been at least implicitly talking about deformations of categories since at least 1994, in the original paper introducing homological mirror symmetry. The idea (or "philosophy") seems to be that the deformation theory of a category should have something to do with its Hochschild (co)homology. But I still do not understand this connection, at least in any sort of generality. Perhaps this is explained in some of the papers already listed in the answers below --- what I'd most like to see is how to relate "deformation of a (linear/dg/A-infinity) category", however one defines it, to Hochschild (co)homology.

Perhaps it's somehow obvious... but I'm pretty dense and would like to see it spelled out...

So I hope someone can explain this to me, or point me to a spot in a paper where it is explained.

I'm adding a bounty to this question just for the heck of it.

Edit 2: See my answer below.

Answer 1

Certainly not my field, but you might want to check the paper by Lowen and Van den Bergh Deformation theory of abelian categories. I think that's where the first notion of deformation of a category appeared.

Answer 2

Here is a definition that I heard: a family of additive categories over a scheme $X$ (say, defined over a field $k$) is an $k$-linear additive category ${\mathcal C}$ equipped with a structure of a module category over the monoidal category of (quasi)coherent sheaves on $X$. Apparently, this setting is rather general and flexible, e.g. allows one to make sense of the notion of flatness of such a family, etc.

I also want to mention that there is now a new paper of van den Bergh on this, arXiv:1002.0259.

Answer 3

Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the category of coherent sheaves on $X$. These deformations are classified by the second Hochschild cohomology of $X$, which by the Hochschild-Kostant-Rosenberg theorem is $H^0(\wedge^2T)\oplus H^1(T)\oplus H^2({\mathcal O})$, where ${\mathcal O}$ is the structure sheaf and $T$ is the tangent sheaf of $X$.

Here, roughly speaking, the first summand corresponds to noncommutative deformations of the structure sheaf (global Poisson bivectors), the second summand corresponds to commutative deformations of this sheaf (i.e., formal deformations of X as a variety), and the third term corresponds to deformations of the category of coherent sheaves which do not arise from deformations of the structure sheaf as a sheaf of algebras (i.e., the algebra deformations exist only on local charts but do not glue into a sheaf; only their categories of modules glue into a sheaf of categories, called a gerbe).

Another helpful reference on this may be van den Bergh's paper arXiv:math/0603200.

Answer 4

You can think of small additive category over a field $F$ nothing more than an associative algebra with distinguished idempotents. Here a "small" category is one in which the classes of objects and morphisms are small enough to be a set. An "additive" category over a field is one in which the hom spaces are all vector spaces over $F$, but we do not demand the extra structure in an abelian category, namely kernels or cokernels or direct sums. To turn the category into a ring, take the direct sum of all of the hom spaces. The identity elements of the objects become distinguished idempotents. (The construction is a little more reasonable if the category is skeletal, i.e., if there is only one object of each isomorphism type.)

The projections satisfy some properties. All pairs of them multiply to zero, and they serve as a partitioned identity element for the algebra. (If there are infinitely many objects, then the ring does not have an identity element.)

For example, the path algebra of a quiver is traditionally described as an algebra, but it is really a category described in exactly this way, with one object for each vertex of the quiver.

Once you realize that an $F$-additive category is just an algebra with some marked elements, you can deform it. For instance, the Temperley-Lieb category is the category whose objects are $n$ points on a string, and whose morphisms are linear combinations of planar matchings of $n+k$ points, with $n+k$ even. The circle has a scalar value, and this parameter gives you a one-parameter family of deformations of the TL category.

If you deform an algebra $A$, then you can sometimes say that you are deforming the category of $A$-modules. But sometimes not, because at special values of the parameters, the hom spaces of between the $A$-modules might be larger, and new modules might appear. Just as the dimension of the kernel of a linear map is upper semicontinuous, but not continuous. If you want to be fancy, you could say that the homs make a sheaf on the parameter variety and that the dimension is upper semicontinuous. You could even say in a generalized sheaf sense that it is still a deformation of the category of modules.

Answer 5

Ok, so I finally spent some time looking through some papers of van den Bergh and Lowen in more detail, which is probably what I should have done a while ago.

The first relevant paper to my questions is probably the paper that javier had originally posted, on deformations of abelian categories: http://arxiv.org/abs/math/0405226

Then there is a sequel paper http://arxiv.org/abs/math/0405227 which claims to explain how Hochschild cohomology is related to deformations of abelian categories! This looks to be exactly what I wanted, or at least close enough... However, this paper does not seem to have any concrete examples...

There are other papers of van den Bergh and Lowen that are probably relevant and that I should probably look at further...

Answer 6

Without looking over it again, I think Seidel's "Fukaya categories and deformations" http://arxiv.org/abs/math/0206155 would be in some sense both an example and a bit of general theory. And only 9 pages, too.

I seem to recall a talk by Seidel where the deformation of $A_\infty$ category was controlled by appropriate Hochschild cohomology group -as Greg points out, small categories are just algebras over semisimple rings, and deformations of algebras are controlled by Hochschild groups (well, classically not of $A_\infty$ algebras, but it extends (though I have not personally checked)). Then there was the point that in the particular example some of the deformations were geometric (think deforming complex structure for DCoh, or symplectic form for DFuk) and some were not. Some of the "not" ones were appropriately viewed as coming from non-commutative deformations of the underlying manifold. I wish I remembered more details. Maybe I can edit this answer if I find notes...

Edit: After looking it over: Section 5 in Seidel's paper looks to indeed be (prtial?) answer to your question. The referenced paper of Fukaya I have not read, but it looks very interesting (in general, and with regards to this question in particular).

Answer 7

Our paper might be of interest: http://arxiv.org/abs/math/0509161 in regards to non-commutative and gerbey deformations. If you have a deformation of a category over some paramerter space $S$, the Hom spaces should become spaces over $S$ as mentioned by G.K. You can ask if an object from the original category "deforms". It might not, in other words it might not be the reduction to the central fiber of some object over $S$.

Answer 8

This is treated in the thesis of Mathieu Anel, downloadable here. In part IV.3.2 he explains, in the framework of derived algebraic geometry, What is the tangent complex of the moduli stack of Abelian Categories - it is something quasiisomorphic to a 2-truncated Hochschild complex...

Answer 9

Here's a slide talk that might be of some help.

www.math.uwo.ca/~ewu22/uwoalgebraseminar.pdf