The regular representation of an essentially finite 2-group G in the 2-category 2Vectk of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in Rep 2Vectk pGq are 2-vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers” is obtained wh...

1.1. Motivation. Over C and over non-archimedean fields, analytification of algebraic spaces is defined as the solution to a quotient problem. Such analytification is interesting, since in the proper case it beautifully explains the essentially algebraic nature of proper analytic spaces with “many” algebraically independent meromorphic functions. (See [A] for the complex-analytic case, and [C3]...

We want F to satisfy some kind of Mayer-Vietoris property, or excision. Hence, we assume C and D are proper, in that the pushout of a weak equivalence along a cofibration is also a weak equivalence. We’ll also ask that in D, sequential colimits of homotopy Cartesian cubes are again homotopy Cartesian, and we’ll elaborate on what this means. We also place a condition on F : Goodwillie calls it “...

Given a monad T on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category,...

This paper describes one approach for calculating the cohomology of a particular complex which arises in knot theory and which is used to prove the existence of ‘associators’, a key ingredient in the construction of a universal knot invariant. The approach taken is to show that the relevant complex can be interpreted, essentially, as the cohomology of the functor Hom(K, C), where C is the coalg...

Using one of Wodzicki's examples of H-unital algebras [14] we exhibit a ring whose derived category contains a smashing subcategory which is not generated by small objects. This disproves the generalization to arbitrary triangulated categories of a conjecture due 1. Statement of the conjecture We refer to [7] for a nicely written analysis of the following setup: Let S be a triangulated category...

In this paper we present background results in enriched category theory and model necessary for developing categories of functors suitable doing functor calculus.

It is known that Plotkin’s reduction theorem is very important for his theory of universal algebraic geometry [1, 2]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This new proof and applications are the subject of the present paper. INTRODUCTION An automorphism φ of a categor...

The plausibility and desirability of such a functor E were shown by Adams [2]. To obtain an existence proof (§3), I will construct an appropriate localization functor on the category of simplicial sets and will show that it induces the desired h,-localization functor on Ho. The backbone of this proof is in an appendix ($10~$12), where I introduce a version of simplicial homotopy theory in which...