Sergey Arkhipov

2-Gerbes and 2-Tate Spaces

We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbal theories. This central extension is used to define central extensions of double loop groups.

Semicontramodules and Semihomomorphisms

Throughout Chapters 3-11, k∨ is an injective cogenerator of the category of k-modules. One can always take k∨ = HomZ(k,Q/Z).

Semialgebras and Semitensor Product

Throughout Chapters 1-11, k is a commutative ring. All our rings, bimodules, abelian groups, … will be k-modules; all additive categories will be k-linear.

Preliminaries and Summary

This chapter contains some known results and some results deemed to be new, but no proofs. Its goal is to prepare the reader for the more technically involved constructions of the main body of the monograph (where the proofs are given). In particular, we do not have to worry about nonassociativity of the cotensor product and partial definition of the semitensor product here, distinguish between the myriad of notions of absolute/relative coflatness/coprojectivity/injectivity of comodules and analogously for contramodules, etc., because we only consider coalgebras over fields.

Functoriality in the Semialgebra

Let ϱ → D be a map of corings compatible with a k-algebra map A → B. Let S be a semialgebra over the coring ϱ and T be a semialgebra over the coring D.

Derived Functor SemiExt

Let A be an exact category in which all infinite products exist and the functors of infinite product are exact. A complex C• over A is called Italic if it belongs to the minimal triangulated subcategory Acycl ctr (A) of the homotopy category Hot(A) containing all the total complexes of exact triples ′K• → K• → ″K• of complexes over A and closed under infinite products. Any contraacyclic complex is acyclic. It follows from the next lemma that any acyclic complex bounded from above is contraacyclic.

Closed Model Category Structures

By a closed model category we mean a model category in the sense of Hovey [49]. The closed model categories that we will construct are also abelian model categories in the sense of [51], so our results can be viewed as particular cases of the general framework developed in [50].

Relative Nonhomogeneous Koszul Duality

All the constructions of Chapters 1-10 can be carried out with the category of k-modules replaced by the category of graded k-modules.

Derived Functor SemiTor

A complex C• over an exact category [28] A is called exact if it is composed of exact triples Zi → Ci → Zi+1 in A. A complex over A is called acyclic if it is homotopy equivalent to an exact complex (or equivalently, if it is a direct summand of an exact complex). Acyclic complexes form a thick subcategory Acycl(A) of the homotopy category Hot(A) of complexes over A. All acyclic complexes over A are exact if and only if A contains images of idempotent endomorphisms [69].