# Graduate Category Theory Seminar

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Currently the category theory seminar is organized by Patrick Nicodemus, nicodemus@math.wisc.edu. There is a mailing list, join-categorytheory@lists.wisc.edu, which sends out announcements on about a weekly basis. In person meetings are cancelled indefinitely due to coronavirus.

## Spring 2018

- Week One - Plan of seminar established. Definition of Kan extension; colimit formula for the Kan extension. Following Mac Lane's Categories for the Working Mathematician.
- Week Two - Properties of Kan extensions.
- Week Three - Coends, examples of coends. The coend formula for the Kan extension.
- Week Four - Definition of a simplicial set. The Density theorem, phrased in terms of Kan extensions but illustrated in the case of simplicial sets.
- Week Five - the "nerve-realization adjunction" theorem (which occurs as I.5.2 in Mac Lane and Moerdijk, "Sheaves in Geometry and Logic") phrased in terms of the left Kan extension of the functor A : C -> E along the Yoneda embedding. Some examples of the theorem: the adjunction between presheaves over X and Top/X, the adjunction between the singular total complex functor from simplicial sets.
- Week Six - Spring break.
- Week Seven - First online session due to Coronavirus. The Nerve of a Small Category and the Fundamental Groupoid of a Simplicial Set, available here.
- Week Eight (Planned) - Geometry of simplicial sets. Whence degeneracies? The Eilenberg-Zilber theorem. Acyclic Models.