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

A Type Theory for Strictly Unital ∞\infty-Categories

We present a type theory for strictly unital $\infty$-categories, in which a term computes to its strictly unital normal form. Using this as a toy model, we argue that it illustrates important unresolved questions in the foundations of type theory, which we explore. Furthermore, our type theory leads to a new definition of strictly unital $\infty$-category, which we claim is stronger than any previously described in the literature.Comment: 45 page

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

Types are internal infinity-groupoids

International audienceBy extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of ∞-groupoid internal to type theory and we prove that the type of such ∞-groupoids is equivalent to the universe of types. That is, every type admits the structure of an ∞-groupoid internally, and this structure is unique

The Nilpotence Tower

Much like the theory of affine schemes and commutative rings, the theory of (higher) topoi leads a dual life: one algebraic and one geometric. In the geometric picture, a topos is a kind of generalized space whose points carry the structure of a category. Dually, in the algebraic point of view, a topos may be thought of as the "ring of continuous functions on a generalized space with values in homotopy types". In this talk, I will explain the connection between Goodwillie's calculus of functors and this algebro-geometric picture of the theory of higher topoi. Specifically, I will describe how one can view the topos of n-excisive functors as an analog of the commutative k-algebra k[x]/x⠿⠺¹, freely generated by a nilpotent element of order n+1. More generally, I will show how every left exact localization E â F of topoi may be extended to a tower of such localizations E â ¯ â Fâ â Fâ â â â â ¯ Fâ = F which we refer to as the Nilpotence Tower, and whose values at an object of E may be seen as a generalized version of the Goodwillie tower of a functor with values in spaces. Under the analogy with scheme theory described above, this construction corresponds to the completion of a commutative ring along an ideal, or, geometrically, to the filtration of the formal neighborhood of a subscheme by it's n-th order sub-neighborhood. I will also explain how, in addition to the homotopy calculus, the orthogonal calculus of Michael Weiss can be seen as an instance of this same construction. This is joint work with M. Anel, G. Biedermann and A. Joyal.Non UBCUnreviewedAuthor affiliation: University of CambridgePostdoctora