## Homotopy cyclic 2-category

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 on the operad of chains of little discs acts. Below a notion of homotopy cyclic 2-category is introduced. For it the complex is acted by the operad of chains of *framed* little discs.

DG 2-category is defined by the following data

- a set of objects
- a set of 1-morphisms for
- a complex of 2-morphisms for any for , where .

For any set of objects and set of 1-morphisms a morphism (composition)

is given, where and . 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

where 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

isomorphisms

are given, compatible with composition morphisms.

One may easily see, that on 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 objects are points of , 1-morphisms between and are paths from to and complex of 2-morphisms is the chain complex of the space of homotopies between paths. The cyclic structure is obvious. The complex is the chain complex of the double loop space . 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.

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