Strictification of $(\mathrm{\infty },1)$-Categories.

$\mathrm{\infty }$-groupoids are infinite-dimensional generalizations of groupoids in which all axioms (associativity, identity, etc.) hold only “up to (coherent) homotopy.” This flexibility makes them powerful enough to encode homotopy types, but much too complicated to reason with in any algebraic manner. $\omega$-groupoids are a “strict” version, in which all the relevant axioms hold on the nose, rather than up to homotopy. There is an adjunction

In which the left adjoint $\mathrm{St}$ “strictifies” an $\mathrm{\infty }$-groupoid, forcing axioms to hold strictly rather than simply up-to-homotopy. Since $\mathrm{\infty }$-groupoids are equivalent to homotopy types, $\mathrm{St}$ gives an algebraic invariant for spaces. It turns out that $\omega$-groupoids are essentially a blend of chain complexes and (ordinary) groupoids, and the functor $\mathrm{St}$ refines simultaneously the homology and fundamental groupoid of a space. Various homological theorems that hold only for simply connected spaces can be expanded to all spaces by working with $\omega$-groupoids rather than chain complexes: for instance, the functor $\mathrm{St}$ reflects weak equivalences, which specializes to simply connected spaces as the statement that a map $f:X\to Y$ is a weak equivalence of simply connected spaces iff it is a homology equivalence.

In this talk, we will discuss the construction of an analagous invariant for $(\mathrm{\infty },1)$-categories: we construct a Quillen adjunction

Where $(\omega ,1)\mathrm{Cat}$ is the category of $(\omega ,1)$-categories, which are “strict” versions of $(\mathrm{\infty },1)$-categories [2]. We show that while essentially a mix of chain complexes and (strict) $(2,1)$-categories, this invariant is fairly strong: in particular, we show that ${\mathrm{St}}_1$ reflects weak equivalences, thus giving a homological/categorical mechanism to test for equivalences of $(\mathrm{\infty },1)$-categories [3].

As an important technical step, we prove a change-of-base theorem for monoidal model categories: given a Quillen adjunction $L:𝒱\rightleftarrows 𝒲:R$ with $R$ lax monoidal, we prove that (under some model categorical assumptions) there is an induced Quillen adjunction

Notably, we improve on previous results of this form (e.g. [1]) by not requiring that $L$ be “weak Quillen monoidal,” the homotopical analog of strong monoidality [3].