Nikita Markarian’s mathblog

TQFT as a generalization of spherical homology

Posted in Mathematics by nikitamarkarian on December 15, 2009

Speculating about the question posed in the end of this confused post I recognised, that the generalization involved makes some sense. Spherical homology is generalised in two directions. Firstly 2-sphere is replaced with any compact oriented 2-manifold. Secondly a BV (f2-)algebra is replaced with a cyclic 2-category. As a result one get a structure that strongly reminds TQFT up to decorations. By decorations I mean metric, counit and so on. This conjectural structure plays for cyclic 2-categories the same role as  Hochschild homology plays for (1-)categories.

Let us start with Hochschild homology of a DG category C. The standard complex calculating it is built of \mathop{Hom}(X_0, X_1)\otimes \mathop{Hom}(X_1, X_2)\otimes\cdots\otimes\mathop{Hom}(X_n, X_0) , where X_i are objects of the category (the definition may be easily given for a homotopy [i.e. A_\infty-] category as well). If the category has only one object, then we get the usual Hochschild complex.

This may be rephrased as follows. Take a circle and consider all triangulations of it, that is all sets of finite number of circle points. Then put objects of the category in the 0-cells, that is points, and Hom’s – in 1-cells. It means, take moduli space of n different points of the circle times (\mathop{Ob} C)^n and consider the complex of chains on it with coefficients in the tensor product of Hom’s, corresponding to intervals:

Now glue equivalent triangulations, that is if we remove a 0-cell, then tensor product of Hom’s we replace with the composition:

The resulting complex obviously calculates Hochschild homology of the category. (Here is some cheating, we did not consider all triangulations, it is not clear, what to do when two arrows go towards. But I am sure that the construction may be tuned in a suitable way.)

Let X be a connected topological space. Consider the following DG category. Objects are points of X. The complex of morphisms from x\in X to y\in X is the chain complex of (Moore) paths from x to y. As it follows e.g. from the Goodwillie’s result that we discussed many times, Hochschild homology of this category is the chain complex of the free loop space (S^1, X).

Now let us turn to 2-categories.

At the beginning, the following was not explicitly stated: in the definition of the spherical homology on may replace 2-sphere with any oriented 2-manifold S. Let us call the resulting object by S-homology. For a space X, S-homology of the chain complex of \Omega^2 X is without a doubt the chain complex of the space of maps (S, X).

By the previous post, a categorical analog of a f2-algebra is a homotopy cyclic 2-category. This observation leads to the following conjecture.

Conjecture: For any compact oriented 2-manifold S there exist a functor from (homotopy) cyclic 2-categories to complexes such that for a category with one object and one 1-morphism this is S-homology.

Call this functor by S-homology of a category.

The category of 2-paths was considered in the end of the previous post: for a connected and simply connected topological space X objects are points of X, 1-morphisms between x and y are paths from x to y and complex of 2-morphisms is the chain complex of homotopies between paths. S-homology of this category must be the chain complex of the space of maps (S,X).

Another example is given by the following (linear) 2-category (see e.g. section 8.3 of J.C. Baez, A.D. Lauda “Higher-Dimensional Algebra V: 2-Groups” and references therein). It has one object, 1-morphisms form a group G and all 2-morphisms are only from a 1-morphism to itself and one-dimensional. This category is defined by the 3rd cohomology of G in the multiplicative group of the base field. For a finite group this example is connected with Cher-Simons theory for finite groups, see R. Dijkgraaf, E. Witten, “Topological gauge theories and group cohomology”. If G is an algebraic group (e.g. SL_n) and the cocylce is the standard one, that is the second Chern character, this example is connected with the Chern-Simons TQFT, see the mentioned paper of Dijkgraaf-Witten and also series of papers D.S. Freed, M.J. Hopkins, C. Teleman, “Loop groups and twisted K-theory”.

How does a definition of S-homology of a category could look like? The only idea I have is to consider the space (more exactly, the limit of spaces) of all triangulations of S with vertices marked by objects of the category and edges marked by 1-morphisms and take cohomology of the local system (limit of local systems) formed by 2-morphisms. More manageable definition could include surfaces with boundaries, as in TQFT. For a surface with n border circles the functor should give a complex of (twisted) sheaves over HH(M)^{\times n}, where HH(M) is Hochschild homology of the 2-category considered as a (1-)category and by “twisted” I mean twisting by an Abelian gerbe. If 1-morphisms form a set, then HH(M) is a simplicial set, but if 1-morphisms are equipped with some additional structure (e.g. it is a scheme), then this set inherits it, and twisted sheaves goes with an additional stricture (e.g. they are twisted sheaves of O-modules). The picture should be supplemented by gluing rules.

Continuing the flight of fancy, one may associate a metric with any triangulation, then consider the projection from the space of all triangulations to the moduli space of conformal classes of metrics and take the direct image. In this way one get a complex over this moduli space.


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