On localizations via homotopies

DUBUC, EDUARDO J

Conferencia; Topos Institute Colloquium; 2022

Let $\mathcal{C}$ be a category and $\Sigma$ be a class of morphisms. The \emph{localization} of $\mathcal{C}$ at $\Sigma$ is a category $\mathcal{C}[\Sigma^{-1}]$ together with a functor $q: \mathcal{C} \longrightarrow \mathcal{C}[\Sigma^{-1}]$ such that $q(s)$ is an isomorphism for all $s \in \Sigma$, and which is initial among such functors. The \emph{2-localization} is a 2-category $\mathcal{C}[\Sigma^{\sim 1}]$ together with a functor $q: \mathcal{C} \longrightarrow \mathcal{C}[\Sigma^{\sim 1}]$ such that $q(s)$ is a equivalence for all $s \in \Sigma$, and which is initial among such functors. In this talk I will consider the construction of such localizations by means of cylinders and its corresponding homotopies, which will determine the \mbox{2-cells} of $\mathcal{C}[\Sigma^{\sim 1}]$. I will examine the case where $\Sigma = \mathcal{W}$ is the class of weak equivalences of a Quillen's model category, and in particular the role of the fibrant and cofibrant replacements. I will elaborate about functorial versus non functorial factorizations in this construction. I will recall informally but with sufficient precision the necessary definitions so that the non experts can grasp the ideas and follow the proofs.