Operation-indexing categories and Grothendieck fibrations

Edouard Balzin – 14 August 2019

It has been known since Segal that various small categories can be used as blueprints for algebraic structures in homotopy theory, providing alternatives to operads in such questions as for example delooping. The examples of those categories include finite sets, ordered sets, n-ordinals of Batanin and various exit path categories of configuration spaces, as well as categories of operators of general topological operads. More recently, the Lurie has shown how to develop higher-categorical algebra using the language of various fibrations over such categories, provided one is willing to pay the price of (immense) technical difficulty.

In our talk, we would like to explain the examples of various operation-indexing categories (and offer a definition of such a notion due to Harpaz) and how one can use them to describe homotopy-algebraic structures via the fibrations of model and higher categories. Our approach is arguably much less technical than that of Lurie, and provides what can be viewed as "Segal objects" in chain complexes of vector spaces. Depending on time and the interest of the audience, we may also present the particular example of the category of trees of Kontsevich-Soibelman and discuss its relation to the Hochschild cohomology.