In terms of category theory, the Gromov homotopy principle for a set valued functor F asserts that the functor F can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor F holds if the functor F can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by bordism groups of solutions ...

Informally speaking, a map is stratified if it is proper and there is a partition S of X into manifolds so that over each manifold S ∈ S the map f(S)→ S is a fiber bundle. One perspective on this question uses the open sets of X to define a functor F̂ : Open(X)→ Set that assigns to each open U the set π0(f (U)) of path components. This functor satisfies a gluing axiom reminiscent of the usual va...

We prove in this note that the global geometric theta lifting for the pair (H,G) is compatible with the Whittaker normalization, where (H,G) = (SO2n, Sp2n), (Sp2n, SO2n+2), or (GLn, GLn+1). More precisely, let k be an algebraically closed field of characteristic p > 2. Let X be a smooth projective connected curve over k. For a stack S write D(S) for the derived category of étale constructible Q̄...

A 2-group is a ‘categorified’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak’ and ‘coherent’ 2-groups. A weak 2-group is a weak monoidal category in whi...

Applicative functors [6] are a generalisation of monads. Both allow the expression of effectful computations into an otherwise pure language, like Haskell [5]. Applicative functors are to be preferred to monads when the structure of a computation is fixed a priori. That makes it possible to perform certain kinds of static analysis on applicative values. We define a notion of free applicative fu...

The goal of this paper is to provide the last equivalence needed in order identify all known models for ( ∞ , 2 ) $(\infty,2)$ -categories. We do by showing that Verity's model saturated 2-trivial complicial sets equivalent Lurie's $\infty$ -bicategories, which, turn, has been shown be other A key technical input given identifying notion -bicategories with weak a step which allows us understand...

We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their functors. Our methods seem to exhibit new anabelian-type phenomena, in the form of structures in the category of schemes that encode automorphism data in gro...

We introduce and discuss the notion of naturally full func-tor. The definition is similar to the definition of separable functor: a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study general properties of naturally full functors. We also discuss when func-tors between module categories and between c...

in this paper we show that expansion of a buchsbaum simplicial complex is $cm_t$, for an optimal integer $tgeq 1$. also, by imposing extra assumptions on a $cm_t$ simplicial complex, we provethat it can be obtained from a buchsbaum complex.