Varieties of trees.

Abstract: In this lecture we will present various categories of trees, together with Quillen model structures on the associated categories of simplicial presheaves. The central example is formed by the category Omega and the complete Segal model structure on its simplicial presheaves, which provides a model for the homotopy theory of infinity-operads analogous to (and in fact, in a strict sense, containing) Rezk's complete Segal model for infinity categories. But unlike this simplicial case, the case of trees allows for many variations of the underlying category of trees as well as of the model structure. To mention a few variations, one obtains in this way model categories for the homotopy category of unital operads, that of algebras over a given infinity-operad, and that of infinite loop spaces, for example.

The lecture will survey these topics in a hopefully generally accessible way. Later in the programme, we will go into some more technical aspects of the theory.

Reference: G. Heuts, I. Moerdijk, Trees in Algebra and Geometry – An introduction to dendroidal homotopy theory (draft of book available on our websites).

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)