# The Gray tensor product for 2-quasi-categories

thesis

posted on 2022-03-29, 01:44 authored by Yuki MaeharaThe content of this thesis is intended as a steppingstone towards reconstructing Street's formal theory of monads [Str72] in the (∞,2) context. Although it is not explicitly mentioned in Street's original paper, the formal theory makes use of the monoidal closed structure on the category 2-Cat given by the (lax) Gray tensor product [Gra74]. More specifically, it requires the 2-category of 2-functors, lax natural transformations and modifications (which is the left closed part of this structure) since Street characterises the familiar Eilenberg-Moore category of algebras as the lax limit of the monad in an appropriate sense. This thesis demonstrates that a homotopical counterpart of this monoidal closed structure exists. A more precise formulation is given at the end of this summary. We adopt 2-quasi-categories, which are the fibrant objects in » op 2 ; Set¼ with respect to a model structure due to Ara [Ara14], for modelling (∞,2) categories. In that paper, Ara characterised not only the 2-quasi-categories, but also the fibrations into them. More precisely, he proved them to be exactly those maps with the right lifting property with respect to a set JA of monomorphisms. The purpose of Chapter 3 is to provide an alternative to JA that is better suited for our purposes, i.e. combinatorics. More precisely, we prove that the set JO consisting of Oury's inner horn inclusions and equivalence extensions [Our10] can be used in place of JA. In Chapter 4, we construct the 2-quasi-categorical Gray tensor product extending the 2-categorical one in an appropriate sense. Although this tensor product is not associative up to isomorphism, we can define the n-ary tensor product for each n 0 and organise them into a lax monoidal structure on » op 2 ; Set¼. That is, there exist appropriately coherent, but not necessarily invertible, comparison maps from nested tensor products to the corresponding total tensor products, e.g. 2¹ 2¹X;Yº; Zº ! 3¹X;Y; Zº. We then use the combinatorial tool developed in Chapter 3 to prove that this lax monoidal structure may be regarded as a genuine monoidal (closed) structure in a homotopical sense. More precisely, each n-ary tensor product functor is shown to be left Quillen with respect to Ara's model structure, and also the (relative) comparison maps are shown to be trivial cofibrations.