We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the Bisson-Tsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories.

Les travaux présentés dans cette thèse concernent l'étude des représentations et la cohomologie des foncteurs strictement polynomiaux d'une ou plusieurs variables. Ces foncteurs d'une variable ont été introduits par Friedlander-Suslin dans leur article fondamental [Eric M. Friedlander and Andrei Suslin. Cohomology of finite group schemes over a field. Invent. Math., 127(2) : 209-270, 1997]. Des variantes à plusieurs variables sont définies dans [Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin. General linear and functor cohomology over finite fields. Ann. of Math (2), 150(2) : 663-728, 1999], [Vincent Franjou and Eric Friedlander. Cohomology of bifunctors. Proc. Lond. Math. Soc. (3), 97(2) : 514-544, 2008], [Antoine Touzé. Cohomology of classical algebraic groups from the functorial viewpoint. Adv. Math., 225(1) : 33-68, 2010]. Dans une première partie de la thèse (chapitre 2), nous étudions l'influence de la torsion de Frobenius sur les groupes d'extension dans les catégories des foncteurs strictement polynomiaux. [...] / The work presented in this thesis deals with the study of representations ans cohomology of strict polynomial functors. The category P* of strict polynomial functors was introduced by Friedlander and Suslin in thei fouder article [Eric M. Friedlanderand Andrei Suslin. Cohomology of finite group schemes over a field. Invent. Math., 127(2) : 209-270, 1997]. Variants of this category, that is strict polynomial functors with several variables were introduced in [Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin. General linear and functor cohomology over finite fields. Ann. of Math (2), 150(2) : 663-728, 1999], [Vincent Franjou and Eric Friedlander. Cohomology of bifunctors. Proc. Lond. Math. Soc. (3), 97(2) : 514-544, 2008], [Antoine Touzé. Cohomology of classical algebraic groups from the functorial viewpoint. Adv. Math., 225(1) : 33-68, 2010]. The first part of this thesis (chapter 2) studies the effect of Frobenius twist on the cohomology of strict polynomial functors. [...]