Title:
Rudiments of Homotopy Combinatorics
Abstract:
Where classical combinatorics deals with finite sets of
structures, homotopy combinatorics deals with finite homotopy
types of structures. The basic notion is that of homotopical
species. I will explain two theorems: the first (joint work
with David Gepner) is a 'Joyal theorem' for homotopical species,
characterising their associated analytic functors in terms of
exactness conditions. The second (joint work with Imma Galvez
and Andy Tonks), is a 'Schmitt theorem', to the effect that
restriction species give rise to incidence coalgebras, via the
notion of decomposition space. Motivation comes on one hand
from program semantics (generic datatypes), and on the other
hand from quantum field theory (Feynman graphs). In both cases
the need for a homotopical setting comes from the presence of
symmetries. (1-groupoids would actually be enough to deal with
these examples, but it is practical to develop the theory in
the setting of infinity-groupoids.)