Dimensional reduction and the equivariant Chern character

We propose a dimensional reduction procedure in the Stolz--Teichner framework of supersymmetric Euclidean field theories (EFTs) that is well-suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern character for manifolds with an action by a finite group.Comment: 29 pages. Exposition improvements and expanded appendix

Derived Algebraic Geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science

Sets in homotopy type theory

Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi W$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets

Concrete fibrations

As far as we know no notion of concreteness for fibrations exists. We introduce such a notion and discuss some basic results about it

What should a generic object be?

Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of generic object and split generic object, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $\lambda2$-fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out many fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in the sense of Jacobs. We argue for a new alignment of terminology that emphasizes the forms of generic object that appear most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphic types. In addition, we propose a new class of acyclic generic objects inspired by recent developments in the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration

Fibrational linguistics: Language acquisition

In this work we show how FibLang, a category-theoretic framework concerned with the interplay between language and meaning, can be used to describe vocabulary acquisition, that is the process with which a speaker $p$ acquires new vocabulary (through experience or interaction). We model two different kinds of vocabulary acquisition, which we call `by example' and `by paraphrasis'. The former captures the idea of acquiring the meaning of a word by being shown a witness representing that word, as in `understanding what a cat is, by looking at a cat'. The latter captures the idea of acquiring meaning by listening to some other speaker rephrasing the word with others already known to the learner. We provide a category-theoretic model for vocabulary acquisition by paraphrasis based on the construction of free promonads. We draw parallels between our work and Wittgenstein's dynamical approach to language, commonly known as 'language games'.Comment: ACT2022 version; FibLang chapter 0 is at arXiv:2201.0113