Grothendieck topologies from unique factorisation systems

This work presents a way to associate a Grothendieck site structure to a category endowed with a unique factorisation system of its arrows. In particular this recovers the Zariski and Etale topologies and others related to Voevodsky's cd-structures. As unique factorisation systems are also frequent outside algebraic geometry, a construction applies to some new contexts, where it is related with known structures defined otherwise. The paper details algebraic geometrical situations and sketches only the other contexts.Comment: version 2, completed some proofs, change some reference

Goodwillie's Calculus of Functors and Higher Topos Theory

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for publication by the Journal of Topolog

Shifted symplectic reduction of derived critical loci

We prove that the derived critical locus of a $G$-invariant function $S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red}:X/G\to\mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections.Comment: 26 Pages. Many diagram

Higher Sheaves and Left-Exact Localizations of ∞\infty-Topoi

We propose a definition of higher sheaf with respect to an arbitrary set of maps $\Sigma$ in an $\infty$-topos $\mathcal{E}$. We then show that the associated reflection $\mathcal{E} \to {\rm Sh}(\mathcal{E},\Sigma)$ is left-exact so that the subcategory of sheaves with respect to $\Sigma$ is itself an $\infty$-topos. Furthermore, we show that the reflection $\mathcal{E} \to {\rm Sh}(\mathcal{E},\Sigma)$ may be characterized as the left-exact localization generated by $\Sigma$. In the course of the proof, we study the interaction of various types of factorization systems, and make essential use of the notion of a \emph{modality}, that is, a factorization system whose left class is stable by base change.Comment: 44 page

Grothendieck topologies from unique factorisation systems

This article presents a way to associate a Grothendieck site structure to a category endowed with a unique factorisation system of its arrows. In particular this recovers the Zariski and Etale topologies and others related to Voevodsky's cd-structures. As unique factorisation systems are also frequent outside algebraic geometry, the same construction applies to some new contexts, where it is related with known structures de ned otherwise. The paper details algebraic geometrical situations and sketches only the other contexts

Champs de modules des catégories linéaires et abéliennes

Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations (\ukcatiso, \ukcateq, \ukcatmor) together with the classifying stack of abelian categories (\ukab), the originality of the subject being that, apart from the first one, these are higher stacks.The principal result is that, under some finiteness assumptions, these stacks are geometric in the sense of C.~Simpson. In particular, one recover the Hochschild cohomology of a category $C$ as the tangent complex, i.e. the object classifying first order deformations of $C$, of these stacks at the point defined by $C$.Moreover, there exists a natural sequence of surjective morphisms of stacks :\ukcatiso \tto \ukcateq \tto \ukcatmor \tto \ukabfor which we prove that the middle one is etale, and the right one is an equivalence.Les catégories linéaires ont naturellement plusieurs notions d'identification : l'isomorphie, l'équivalence de catégories et l'équivalence de Morita. On construit les champs classifiant les catégories pour ces trois structures (\ukcatiso, \ukcateq, \ukcatmor) ainsi que le champ classifiant les catégories abéliennes (\ukab), l'originalité étant que les trois derniers champs sont des champs supérieurs.Le résultat principal de la thèse est que, sous des conditions de finitude des objets classifiés, ces champs sont géométriques au sens de C.~Simpson. En particulier, on trouve que les complexes tangents de ces champs en une catégorie $C$, i.e. les objets classifiant les déformations au premier ordre de $C$, sont donnés par des tronqués du complexe de cohomologie de Hochschild de $C$.En plus, il existe une suite naturelle de morphismes surjectifs de champs :\ukcatiso \tto \ukcateq \tto \ukcatmor \tto \ukabdont on montre que celui du milieu est étale, et celui de droite une équivalence