Nikita Markarian’s mathblog

Homotopy cyclic 2-category

Posted in Mathematics by nikitamarkarian on December 9, 2009

In the paper “What do DG categories form?” D. Tamarkin introduced the notion of a homotopy 2-category. Its important feature is that for any object $X$ on $\mathop{Hom}(Id_X, Id_X)$ the operad of chains of little discs acts. Below  a notion of homotopy cyclic 2-category is introduced. For it the complex $\mathop{Hom}(Id_X, Id_X)$ is acted by the operad of chains of framed little discs.
DG 2-category $M$ is defined by the following data

1. a set of objects $\mathop{Ob} M$
2. a set of 1-morphisms $\mathop{Hom}(X,Y)$ for $X, Y\in\mathop{Ob} M$
3. a complex of 2-morphisms $\mathop{Hom}(F,G)$ for any $F,G\in \mathop{Hom}(A,B)$ for $A,B \in\mathop{Hom}(X,Y)$, where $X,Y \in \mathop{Ob} M$.

For any set of objects $X_i$ and set of 1-morphisms $F_{i1}, \dots F_{ik_i}\in\mathop{Hom} (X_i, X_{i+1})$ a morphism (composition)

$\bigotimes \mathop{Hom} (F_{ij}, F_{i,j+1}) \to \mathop{Hom}(F,G)$

is given, where $F=F_{n0}F_{n-1,0}\cdots F_{00}$ and $G=F_{nk_n}F_{n-1, k_{n-1}}\cdots F_{0, k_0}$. These morphisms obey natural relations, for details see the mentioned paper.

A homotopy 2-category is defined by the same data, but the composition is replaced by the morphism

$\O\otimes\bigotimes \mathop{Hom} (F_{ij}, F_{i,j+1}) \to \mathop{Hom}(F,G),$

where $\O$ is a complex. Essentially, this is chains of little discs. For details see ibid.

Definition: A cyclic (homotopy) 2-category is a (homotopy) 2-category with the following additional structure: for any pair of cycles of 1-morphisms

$F_0, G_0\in \mathop{Hom}(X_0, X_1);\dots ;\quad F_n, G_n\in \mathop{Hom}(X_n, X_{n+1}) ;\quad F_{n+1},G_{n+1}\in \mathop{Hom} (X_{n+1}, X_0)$

isomorphisms

$\mathop{Hom} (F_{n+1}F_n\cdots F_0,\quad G_{n+1}G_n\cdots G_0)=\mathop{Hom} (F_{n}F_{n-1}\cdots F_{n+1}, \quad G_{n}G_{n-1}\cdots G_{n+1})=\dots=\mathop{Hom} (F_{0}F_{n+1}\cdots F_n,\quad G_0 G_{n+1}\cdots G_n)$

are given, compatible with composition morphisms.

One may easily see, that on $\mathop{Hom}(Id_X, Id_X)$ in a homotopy cyclic 2-category the operad of chains of framed little discs acts.

As an example consider the following homotopy cyclic 2-category. For a 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 the space of homotopies between paths. The cyclic structure is obvious. The complex $\mathop{Hom}(Id_x, Id_x)$ is the chain complex of the double loop space $\Omega^2 X$. It is acted by the operad of chains of framed little discs, that is not suprising and was already discussed.

UPD: It seems that the reasonable way to introduce the cyclic structure on a homotopy 2-category is to define an involution – reversing – on 1-morphisms and appropriate compatibility morphisms. One should follow the pattern given by the example above.