Third Mac Lane cohomology

Third Mac Lane cohomology via categorical rings

It is proved that the third Mac Lane cohomology group of a ring R with coefficients in a bimodule B classifies categorical rings having R as the ring of isomorphism classes of objects and B as the bimodule of automorphisms of the neutral object.

Finitary Group Cohomology and Eilenberg-Mac Lane Spaces

We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class LHF with cohomology almost everywhere finitary, then G has an Eilenberg--Mac Lane space K(G,1) which is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G.

On the second cohomology group of a simplicial group

We give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.

关于正合 $\infty$-范畴的 Mac Lane $Q$-构造

On the Mac Lane $Q$-Construction for Exact $\infty$-Categories

本文将 McCarthy 的稳定化构造扩展至正合 $\infty$-范畴，通过为任意函子构造相干链复形，直接推广了计算阿贝尔群稳定同调的 Mac Lane 立方 $Q$-复形。

Universal Toda brackets of ring spectra

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.

第二类Mac Lane（上）同调与Wieferich素数

Mac Lane (co)homology of the second kind and Wieferich primes

本文定义第二类Mac Lane（上）同调，计算整体数环有限局部化的该不变量，揭示其与Wieferich素数无限性的等价关系，并确定Mac Lane上同调的环结构及Adams运算。