Théories homotopiques dans les topos. (Homotopy theories in topoi)
The central theme of the paper is to give a construction of a closed model structure [in the sense of \textit{D. Quillen}, ``Homotopical algebra'', Lect. Notes Math. 43 (1967; Zbl 0168.20903)] on any (Grothendieck) topos \(\mathcal E\), given a fixed class \(\mathcal W\) of morphisms (satisfying certain stability conditions) to play the role of weak equivalences and where the cofibrations are the monomorphisms in \(\mathcal E\). Although inspired by the closed model structure given by \textit{A. Joyal} on the category of simplicial sets (unpublished), it is carried out in any topos \(\mathcal E\), in particular in the topos of simplicial sets of any given topos \(\mathcal S\). The paper is strongly influenced by the work and approach of \textit{A. Grothendieck} but also utilizes some ideas which are typical of elementary treatment of topos theory, after \textit{F. W. Lawvere} and \textit{M. Tierney} (such as ``the Lawvere cylinder'', obtained from ``the Lawvere object'', which is the subobjects classifier). It is not clear, however, that everything in this paper could be carried out entirely in an elementary (first-order) manner, as some set-theoretic constructions are used throughout. A first section is devoted to a review of the basic notions of accessible functors [following \textit{A. Grothendieck} in SGA4, Lect. Notes Math. 269 (1972; Zbl 0234.00007)] and to the ``small object'' type of argument and of the factorization of an arrow associated with it. The main section in the paper is the second one, where from the abstract ``homotopy data'' determined by a cylinder in a topos \(\mathcal E\), the author defines anodyne extensions and in turn a closed model structure on \(\mathcal E\) as desired. It is pointed out that this construction, though ad hoc, does not depend on the choice of a cellular model. The rest of the paper is devoted to some refinements of the axiomatics (for instance in order to determine when is the closed model structure a ``proper'' one), and to discussing the motivating example of simplicial sets in this setting. Some of the results, although stated for the most part for an arbitrary topos, sometimes require that the topos in question ``has enough points'' (in the technical sense of topos theory). The paper will be of interest not just to the regular homotopy theorists, but also to those who use closed model structures in their pursuit of higher-order category theory, as well as to topos-theorists with geometric inclinations.
scientific article; zbMATH DE number 1839048