Topology seminars: 2024-25

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I'll report on joint work in progress with Dan Graves (University of Leeds), which extends old work of Teimuraz Pirashvili.

PROPs are very general objects which encode many types of algebraic structure, including algebras, coalgebras, bialgebras, Hopf algebras, .... Operads are simpler objects which encode algebras (but definitely not bialgebras or Hopf algebras) nicely. I'll explain how one can go about assembling some interesting PROPs from operadic ingredients.

I'll try to give this talk at a level appropriate for all (including, in particular, the new PhD students).

When considering (co)limits of categories, one might ask for (co)cones to only commute up to natural isomorphism, or for universal properties to only hold up to equivalences of categories. In a general bicategory K such universal properties are modelled by the notion of a bicategorical (co)limit, where equations/relations are only ever imposed on data in the highest available dimension. However, these notions can also be modelled up to equivalence via ordinary (co)limits enriched over V= Cat, provided that one restricts their attention to weights that are well behaved with respect to the canonical monoidal model structure on V.

In this talk I will explain how the above story adapts to the setting involving (co)limits *of* two-dimensional categories (or more generally, *in* three-dimensional categories). This involves homotopically well-behaved (co)limits enriched over the base V now given by Lack's monoidal model structure on the category of 2-categories and 2-functors. Running examples for motivation include Kleisli and Eilenberg-Moore constructions for pseudomonads, including for those on monoidal bicategories, as well as strictification constructions on bicategories and pseudo-double categories. This talk is based on my recent preprint.