Domenico Fiorenza Chern character

Chern character

First, let me recall the loop space costruction of $\mathbb{Z}$ from $\mathbb{N}$. First step consists in delooping the monoid $\mathbb{N}$, to get a category with a single object $\mathbf{B}\mathbb{N}$, whose set of morphisms is $\mathbb{N}$. Next one consider the nerve of this category. This can be described in very concrete terms as the simplicial set (actually a simplicial $\mathbb{N}$-module) $\mathbb{N}^n$ with face and degeneracy maps induced by the sum in $\mathbb{N}$ and by insertion of $0$. Then $\mathbb{Z}=\pi_0(N\mathbf{B}\mathbb{N}))$.

One can verbatim repeat this construction with $Vect$ in place of $\mathbb{N}$. Here $Vect$ is the groupoid of finite dimensional complex vector spaces, i.e. we are taking only isomorphisms of vector spaces as morphisms in $Vect$. Then direct sum (in the larger category of vector spaces with all linear maps as morphisms) and insertion of the $0$ vector space endow the collection $\{Vect^n\}$ with a natural structure of simplicial groupoid (this simplicial groupoid naturally lives into a simplicial 2-vector space (in the sense of Kapranov and Voevodsky)). Taking the nerve of each groupoid $Vect^n$ we end up with a bisimplicial object; we’d like to think to the associated diagonal simplicial object as ‘the nerve of the delooping of $Vect$’. With this in mid we’ll write $N(\mathbf{B}Vect)$ to denote this simplicial object.

Question: is this the Waldhausen construction?

Looking at the set $\mathbb{N}$ as to a trivial groupoid, dimension of a vector space is a functor $dim:Vect\to \mathbb{N}$. Since $dim$ is compatible with (direct) sumes and (tensor) products, it induces a morphism of simplicial groupoids $\{Vect^n\}\to \{\mathbb{N}^n\}$ and so, taking diagonals, a morphism of simplicial sets $dim:N(\mathbf{B}Vect)\to N\mathbf{B}\mathbb{N})$. Taking loop spaces we end up with

$\Omega dim: \Omega N(\mathbf{B}Vect)\to \Omega N\mathbf{B}\mathbb{N})$

Question: is the space on the right hand side the spectrum of singular cohomology?

Question: is the space on the left hand side the spectrum of K-theory?

Question: is $\Omega dim$ the Chern character?

---

Let $V \to X$ be a complex vector bundle with connection $\nabla$ and curvature 2-form

The Chern character of $\nabla$ is the inhomogeneous differential form

where on the right we have $j$ wedge factors of the curvature .

---

A very general notion of Chern character is found in Toen preprint: infty-categories monoidales rigides etc. File Toen web prepr dag-loop.pdf. See also Derived categorical sheaves

http://mathoverflow.net/questions/60403/can-chern-class-character-be-categorified

Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel-2007.pdf. Notion of derived cat sheaves, a categorification of the notion of complexes of sheaves of O-modules on schemes (also quasi-coh and perfect versions). Chern character for these categorical sheaves, a categorified version of the Chern char for perfect complexes with values in cyclic homology. Using the derived loop space. “This work can be seen as an attempt to define algebraic analogues of of elliptic objects and char classes for them”. 1. Motivations: Elliptic cohomology, geometric interpretations, chromatic level and n-categorical level, 2-VBs. Maybe the typical generalized CT of chromatic level n should be related to n-cats, more precisely cohom classes should be rep by maps from X to a certain n-stack. Rognes red-shift conjecture: Intuitively saying that the K-th spectrum of a commutative ring spectrum of chrom level n is of chrom level (n+1). More on ell cohom and 2-cats. Idea of categorical sheaves: For X a scheme, should have a symmetric monoidal 2-cat Cat(X) which is a categorification of Mod(X), in the sense that Mod(X) should be the cat of endomorphisms of the unit objects in Cat(X). More details. Notions of secondary cohomology and secondary K-theory. Notion of derived categorical sheaves, more reasonable than nonderived version. Relation between $S^1$-equivariant functions on LX and negative cyclic homology. 2. Categorification of homological algebra and dg-cats. 3. Loop spaces in DAG. More, including relations with variations of Hodge structures. Final remark on algebraic elliptic cohomology. “Algebraic K-theory determines complex topological K-theory”, ref to Walker 2002.

nLab page on Chern character