Third Mac Lane cohomology via categorical rings

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

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.

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。

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。

Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings

We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to $n$-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal ca…

On 0-homology of categorical at zero semigroups

The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg-MacLane homology of semigroups are given.