Hans-Joachim Baues

Cohomology of small categories

In this paper we introduce and study the cohomology of a small category with coefficients in a natural system. This generalizes the known concepts of Watts [23] (resp. of Mitchell [17]) which use modules (resp. bimodules) as coefficients. We were led to consider natural systems since they arise in numerous examples of linear extensions of categories; in Section 3 four examples are discussed explicitly which indicate deep connection with algebraic and topological problems: (1) The category of Z//p2-modules, p prime. (2) The homotopy category of Moore spaces in degree n, n22. (3) The category of group rings of cyclic groups. (4) The homotopy category of Eilenberg-MacLane fibrations. We prove the following results on the cohomology with coefficients in a natural system: (5) An equivalence of small categories induces an isomorphism in cohomology. (6) Linear extensions of categories are classified by the second cohomology group HZ. (7) The group H’ can be described in terms of derivations. (8) Free categories have cohomological dimension 5 1, and category of fractions preserve dimension one. (9) A double cochain complex associated to a cover yields a method of computation for the cohomology; two examples are given. The results (7) and (8) correspond to known properties of the Hochschild-Mitchell cohomology, see [7] and [ 171. In the final section we discuss the various notions of cohomology of small categories, and we show that all these can be described in terms of Ext functors studied in the classical paper [l l] of Grothendieck.

Quadratic functors and metastable homotopy

Quadratic functors lead to the fundamental notion of a quadratic R-module M where R is a ringoid or a ring. We introduce the quadratic tensor product A ⊗RM and the corresponding abelian group HomR(A,M) consisting of quadratic forms. Then we describe new quadratic derived functors of ⊗ and Hom together with applications for homotopy groups of Moore spaces and (co)homology groups of Eilenberg-Mac Lane spaces.

Quadratic categories and square rings

We consider quadratic categories which generalize the classical additive categories. An additive category A is a category for which morphism sets are abelian groups and the composition fg is bilinear, and for which sums exist in A. A quadratic category Q is slightly more general in the sense that morphism sets are groups and the composition fg is linear in g and quadratic in f. This implies that morphism sets are groups of nilpotency degree 2. We describe below many examples of quadratic categories in algebra and topology which motivate the systematic study of quadratic categories started here; it may be considered as an extension of the investigation of quadratic functors in [4]. The properties of a quadratic category and its subcategories lead to the new notion of a “square ring” which is exactly the quadratic analogue of the classical notion of a “ring”. Indeed each object X in an additive category A yields an endomorphism ring given by all morphisms X → X in A; similarly each object in a quadratic category yields the endomorphism square ring End(X) of X. Hence the connection between square rings and quadratic categories is similar to the relation of rings and additive categories studied by Mitchell [18]. The initial object in the category of rings is the ring Z of integers for which the category of modules is the category of abelian groups. We here determine the initial object Znil in the category of square rings for which the category of modules is the category of groups of nilpotency degree 2. We compute various square rings explicitly, for example, the endomorphism square rings of the suspended projective planes ΣRP2 and ΣCP2. This yields as an application an algebraic description of the homotopy category of all Moore spaces M(V,2) where V is a Z/2-vector space; in fact this category is equivalent to the full category of free objects in the category of 2-restricted nil(2)-groups. There has been recently a lot of interest in operads [9]. In fact, operads O = {On} with On = 0 for n ≥ 3 are the same as special square rings. Therefore the theory of square rings shows naturally how the theory of operads has to be modified in order to deal with nilpotent groups.

The homotopy classification of (n − 1)-connected (n + 3)-dimensional polyhedra, n ≥ 4

THE classification of homotopy types of finite polyhedra is a classical and fundamental task of topology. Here we mean a classification by minimal algebraic data which for example allows the explicit computation of the number of homotopy types with prescribed homology. The main result on this problem in the literature is due to J. H. C. Whitehead (1949) who classified (n I)-connected (n + 2)-dimensional polyhedra. In this paper we consider the next step concerning finite (n I)-connected (n + 3)-dimensional polyhedra. In the stable range, n 2 4. they are classified by the following decomposition thcorcm, see (3.9).

Comparison of MacLane, Shukla and Hochschild cohomologies

It is known that MacLane cohomology coincides with topological Hochschild cohomology ([28]) and coincides also with Baues-Wirsching cohomology of the category of finitely generated free A-modules ([17]). We study the cohomologies mainly in dimensionse 3 since in these dimensions the cohomologies classify various types of extensions. Moreover we are mainly interested in algebras A over a prime field Fp since we apply our results in particular to the Steenrod algebra A over Fp. As one of our main results we prove (see Theorem 7.4.1):

Classification of Abelian Track Categories

We show that each category enriched in Abelian groupoids is a linear track extension and hence is determined up to weak equivalence by a characteristic chomology class. We also discuss compatibility with coproducts. Mathematics Subject Classifications (2000): 18D99, 18G55.

A universal coefficient theorem for quadratic functors

Abstract Let F be a quadratic functor from abelian groups to abelian groups. We compute the derived functors L∗F of F . As an application we compute completely the Quillen homology for the variety of groups of nilpotency degree 2.

Stems and spectral sequences

We introduce the category Pstemanc of n‐stems, with a functor Panc from spaces to Pstemanc. This can be thought of as the n‐th order homotopy groups of a space. We show how to associate to each simplicial n‐stem Q an .nC1/‐truncated spectral sequence. Moreover, if Q D PancX is the Postnikov n‐stem of a simplicial space X , the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X . Similar results are also proven for cosimplicial n‐stems. They are helpful for computations, since n‐stems in low degrees have good algebraic models. 55T05; 18G40, 18G55, 55S45, 55T15, 18G30, 18G10

The algebra of secondary homotopy operations in ring spectra

The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one squares. As an application we obtain new derivations of the homotopy ring.

Third Mac Lane cohomology

Based on the computation of the third author we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups