Title: Homotopy Coherent Adjunctions
Speaker: Dominic Verity
Speaker Info: Macquarie University
Special Note: This is the first part of a two-part series.
Following Joyal's lead, we present adjunctions between oo-categories as adjunctions in an appropriate 2-category; whose unit and counit are homotopy classes of simplicial homotopies. We show that a right or left adjoint can be extended to a simplicially enriched functor whose domain is constructed from the free 2-category containing an adjunction, called the "walking adjunction" by some authors. Our proof includes a very explicit presentation of the data of this "homotopy coherent adjunction." This enriched functor encapsulates both the coherent monad and the coherent comonad generated by such an adjunction.Date: Friday, August 31, 2012
In the second talk, we dive further into the 2-category theory of quasi- categories. In one application, we re-encode the notions of limits and colimits in an ∞-category as absoluting lifting diagrams in the 2-category. This enables immediate, easy proofs of standard categorical results, for instance regarding the preservation of (co)limits by adjoint functors. Another application is an intuitive presentation of the formal theory of monads a la Ross Street. This leads us to a new "formal" proof of the Beck monadicity theorem for adjunctions of ∞-categories.