# Can Chern class/character be categorified?

The Chern character sends the class of a locally free sheaf to the cohomology ring of the underlying variety X. And it is a ring homomorphism from K to H^*. I saw people write its source as the bounded derived category too, which make sense if the underlying variety is smooth (sending a bounded complex to the alternating "sum" of the Chern characters of its cohomology sheaf).

My question is, if I want to think $D^b(X)$ as a certain categorification of $K_0(X)$, is it possible to categorify the chern character map? What will be a good candidate of the target category? (Or is there a heuristic showing this is not likely to be true?)

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There are categorified analogs of the Chern character, but I don't think of them in the way you're proposing. More precisely, you can take an object in the derived category and assign to it a class in cohomology, and this map factors through K-theory, so the two constructions you're discussing seem to me to be the same.

One way to think of the Chern character is the following. Given any associative, dg or $A_\infty$ algebra, you can define its Hochschild homology. This is the recipient for a universal trace map from the algebra, and more generally for any "finite" module (perfect complex) you get a class (its character) in Hochschild homology. Given more generally a (dg or $A_\infty$) category you can similarly define its Hochschild homology and a character map for "finite" objects (which factors through the K-theory of the category), which agrees with the above when your category is modules over an algebra (which it usually is, noncanonically).

To "categorify" you can replace an algebra by an associative algebra object in any symmetric monoidal $\infty$-category. Its Hochschild homology is defined as an object of said category and again there's a Chern character map for "finite" modules. Why is this a categorification? for example you can take your associative algebra to be some derived category of sheaves with a monoidal structure (eg coherent sheaves or $\D$-modules or.. with tensor product or some convolution product), and then its Hochschild homology is itself a category. Thus module categories will have Chern characters which are objects of this homology category. This is (one way to think of) the notion of a "character sheaf" in representation theory (where our associative algebra is sheaves on a group with convolution, and module categories are categories with a nice action of the group, and their Chern character are adjoint-equivariant sheaves on the group, ie categorified class functions).

(This story is by the way a special case of the Cobordism Hypothesis with Singularities of Jacob Lurie -- in fact just of its one-dimensional case.. our algebra objects are assigned to a point, their Hochschild homology is assigned to the circle, modules are allowable "singularities" in the theory and their Chern character is attached to a circle with a marked "singular point")

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David, could you please explain the category structure on Hochschild homology of an associative algebra in a symmetric monoidal $\infty$-category? –  Sasha Apr 3 '11 at 3:59
For the first paragraph above, I understand that I described the same thing, I just want to ask if one can replace the target by a category such that when one goes back one gets the original Chern character. The rest is a bit in over my head, I will try to understand it, though. Thank you for a great reply. –  36min Apr 3 '11 at 4:01
Sasha - I only meant you'll get a category if your algebra itself is one, i.e. lives in an $\infty$-category of categories. For example if you take sheaves on a stack, with tensor product, its Hochschild homology (as well as its Hochschild cohomology, or Drinfeld center) is given by sheaves on the (derived form of the) inertia stack or "derived loop space" (cf. arxiv.org/abs/0805.0157, and arxiv.org/abs/0904.1247 for D-module analogs). –  David Ben-Zvi Apr 3 '11 at 4:24
@36min - that is an interesting point, to which I don't know an answer. I guess I think of passing to characters as a decategorification, as in the case of representations of a group (which is a special case of the discussion). In the usual formulation though we use that this decategorification factors through a map from K-theory to cohomology so it doesn't feel like one! –  David Ben-Zvi Apr 3 '11 at 4:28

The paper http://arxiv.org/abs/0804.1274 of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy.

First of all they introduce a triangulated $2$-category $Dg(X)$ of derived categorical sheaves on a (derived) scheme $X$. It is based on a the idea that a categorification of the theory of modules on a commutative ring $k$ is given by $k$-linear categories: they argue that dg-categories can be used in order to categorify homological algebra in a similar but better way (better in the sens that the non-dg setting seems to be too rigid to allow push-forwards).

The second step is to use, for a given (derived) scheme $X$, the pull-back along the projection $LX\to X$. For a categorical sheaf $F$ on $X$ on consider its pull-back $p^*F$, which naturally come equipped with a self-equivalence $u$. The rough idea to see this is to consider the pull-back (a-k-a >transgression) along the evaluation map $S^1\times LX\to X$, and then to observe that categorical sheaves on $S^1\times LX$ are categorical sheaves on $LX$ together with a $\mathbb{Z}$-action.

Finally, they conjecture the existence of an $S^1$-equivariant trace $Tr^{S^1}(u)\in D^{S^1}(LX)$. Its $K_0$ would be a candidate for the (categorified) Chern character of $F$.

### Why does this categorify the Chern character ?

If we do the same construction starting with a sheaf of $X$, then we get in the end an element in $\pi_0(\mathcal O_{LX}^{S^1})=HC_0^{-}(X)$ (while the non-$S^1$-invariant trace takes values in $\pi_0(\mathcal O_{LX})=HH_0(X)$).

One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace.

### Follow-up

A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in a long paper in french.

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Damien - thanks for the very informative answer! It might be worth pointing out that this construction (for X reasonable - eg a scheme or more generally perfect stack) is a special case of the construction I explain. Namely we consider the dg category QC(X) of quasicoherent sheaves on X, which is an associative (in fact commutative) algebra object in dg-cats. Module categories for it are the same as derived categorical sheaves (or more precisely, quasicoherent ones - alternatively we can work in a sheafified setting from the beginning). –  David Ben-Zvi May 3 '11 at 16:38
The Hochschild homology of QC(X) was calculated in arxiv.org/abs/0805.0157 to be QC(LX) (and the cyclic homology of QC(X) was calculated in arxiv.org/abs/1002.3636 as D-modules on X..for X a scheme - for a perfect stack you get D(LX)). So the Chern character of a derived categorical sheaf is a sheaf on the loop space (and in fact a D-module on X, which is just the cyclic homology of your derived categorical sheaf with its Gauss-Manin connection). In the case of pt/G you recover QC(G/G) ("quasicoherent character sheaves") as the characters of categories with algebraic G-action. –  David Ben-Zvi May 3 '11 at 16:43
(Of course my comment is meant purely mathematically, not historically - the picture I explain certainly owes a great debt to the ideas of Toen and Vezzosi!) –  David Ben-Zvi May 3 '11 at 16:50
Damien - Great! and Yes: an $E_n$ algebra can be integrated on all manifolds of dim at most n via topological chiral homology (ie it's n-dualizable in the appropriate "Morita" higher category). A (left) module for such an algebra is a morphism from the trivial field theory (the simplest kind of example of a "singularity" in the theory), and so we get a "Chern character" for the module which is a class in the chiral homology on any manifold of dim at most n. –  David Ben-Zvi May 3 '11 at 20:51
(If you'd like, this is a statement about functoriality of HH_* for higher algebras: the morphism from the unit corresponding to a module gives a map on chiral homology, and its image defines the Chern character.) –  David Ben-Zvi May 3 '11 at 20:51