Simplicial objects and combinatorial bialgebras

21.01.2022, 11:00  –  Online Seminar
Arbeitsgruppenseminar Analysis

Joachim Kock (University of Barcelona)

The classical constructions of coalgebras and bialgebras from posets, monoids, Möbius categories, operads, and restriction species have a common generalisation to /decomposition spaces/ (Galvez-Kock-Tonks and Dyckerhoff-Kapranov), certain simplicial sets (or simplicial groupoids). I will not get very far into the theory of decomposition spaces in this talk. I will rather take a step back and explain some background for the theory: it will mainly be an introduction to simplicial viewpoints in the context of combinatorial bialgebras, explaining basic simplicial objects such as the classifying space of a monoid, the nerve of a category, and the two-sided bar construction of an operad.
These examples will lead to the notion of decomposition space. Instead of composition, theses structures encode decomposition, and also certain kinds of partial and nondeterministic composition.

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