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 Leonard Guetta committed Aug 24, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 \chapter*{Introduction} The general framework in which the work to be presented in this dissertation takes place is the \emph{homotopy theory of strict $\oo$-categories}, and, as the title suggests, the focus is on homological aspects of this theory. The goal pursued was to study and compare two different homological invariants for strict $\oo$\nbd-categories; that is to say, two different functors $\mathbf{Str}\oo\Cat \to \ho(\Ch)$ from the category of strict $\oo$-categories to the homotopy category of chain complexes in non-negative degre (i.e.\ the localization of the category of chain complexes with respect to quasi-isomorphisms). Before entering the heart of the subject and explaining precisely what the above means, let us immediately mention that, with the only exception of the end of this introduction, all the $\oo$-categories considered will be strict. Hence, we drop the adjective 'strict' and simply say \emph{$oo$-category} instead of \emph{strict $\oo$-category}. Consequently, we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$-categories. The homotopy theory of strict $\oo$-categories most certainly started with the introduction by Street \cite{street1987algebra} of a nerve functor $N_{\omega} : \oo\Cat \to \Psh{\Delta}$ that associates to any $\oo$-category $C$ a simplicial $N_{\oo}(C)$ called the \emph{nerve of $C$} and generalizing the usual nerve of (small) categories in the sense that \iffalse Before entering the heart of the subject and explaining precisely what the above means, let us quickly linger on a terminological detail concerning strict $\oo$-category. In this flavour of higher category theory, the axioms for compositions and units hold strictly, which means that they are witnessed by genuine equalities and not only up to higher dimensional cells. On the other hand, \emph{no} invertibility axioms is required on any dimension. In particular, strict $\oo$-categories are \emph{not} a particular case of $(\infty,1)$-categories in the sense of Lurie (improperly called $\infty$-categories). \fi \iffalse In this vast generalization of category theory, the objects of study have objects and arrows, as for categories, but also arrows between arrows, called $2$-arrows or $2$-cells, arrows between arrows between arrows ($3$-cells), and so on. All these data come equipped with various composition laws between cells. Higher categories come in many different flavours which loosely depend on three main parameters: \begin{itemize}[label=-] \item The maximum dimension of the cells. One then speak of $1$-categories (which is a synonym for the usual categories), $2$-categories, $3$-catgegories and so on. \item The requirement that units and associativity axioms for compositions either are witnessed by genuine equalities \end{itemize} \fi