Online seminar, U. Haifa Topology & Geometry seminar on March 14, 2021

A recording of the talk is available here.

Topology & Geometry Seminar

Speaker: Clemens Berger, University of Côte d’Azur

Topic: Moment categories and operads

Place: This is an online seminar. Please email David Blanc “blanc at math dot haifa dot ac dot il” for the Zoom ID and password.

Time: 16:00

Date: Sunday, March 14, 2021


Almost half a century ago operads have been introduced by May
and Boardman-Vogt. Since then they are used with great success in
different areas of mathematics and even outside. I will present a new
approach based on the concept of moment category.

Starting point is an active/inert factorisation system giving rise to a
family of split idempotent endomorphisms, called moments. Segal’s
category $\Gamma$ plays a universal role here. The inert/active
factorisation system on the dual category of finite sets and partial
maps is the cornerstone of Lurie’s theory of infinity operads.

Moment structures on $\Gamma$, $\Delta$ and $\Theta_n$ encode the
structure of symmetric operad, non-symmetric operad and globular
n-operad (Batanin) respectively. There is an analog of the plus
construction of Baez-Dolan in our setting. It takes a moment category to
a hypermoment category such that operads for the former get identified
with monoids for the latter. These monoids are presheaves satisfying
certain Segal conditions strictly. We show that the plus construction of
Segal’s $\Gamma$ is closely related to the dendroidal category $\Omega$
of Moerdijk-Weiss.