Topological finiteness properties of monoids. Part 1: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

Gr\"obner-Shirshov bases for categories

In this paper we establish Composition-Diamond lemma for small categories. We give Gr\"obner-Shirshov bases for simplicial category and cyclic category.Comment: 20 page

Quillen homology for operads via Gr\"obner bases

The main goal of this paper is to present a way to compute Quillen homology of operads. The key idea is to use the notion of a shuffle operad we introduced earlier; this allows to compute, for a symmetric operad, the homology classes and the shape of the differential in its minimal model, although does not give an insight on the symmetric groups action on the homology. Our approach goes in several steps. First, we regard our symmetric operad as a shuffle operad, which allows to compute its Gr\"obner basis. Next, we define a combinatorial resolution for the "monomial replacement" of each shuffle operad (provided by the Gr\"obner bases theory). Finally, we explain how to "deform" the differential to handle every operad with a Gr\"obner basis, and find explicit representatives of Quillen homology classes for a large class of operads. We also present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin-Vilkovisky algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint arXiv:0912.4895. Final version, to appear in Documenta Mat

Cohomology of Finite Groups: Interactions and Applications (hybrid meeting)

The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the fifth in a series that has emphasized the interactions of group cohomology with other areas. In spite of the Covid-19 epidemic, this hybrid meeting ran smoothly with about half the participants physically present and the other half participating via Zoom

Free Resolutions via Gröbner Bases

In many different settings (associative algebras, commutative algebras, operads, dioperads), it is possible to develop the machinery of Gröbner bases; it allows to find a “monomial replacement” for every object in the corresponding category. The main goal of this article is to demonstrate how this machinery can be used for the purposes of homo-logical algebra. More precisely, we define combinatorial resolutions in the monomial case and then show how they can be adjusted to be used in the general homogeneous case. We also discuss a way to make our monomial resolutions minimal. For associative algebras, we recover a well known construction due to Anick. Various applications of these results are presented, including a new proof of Hoffbeck’s PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operad of Batalin–Vilkovisky algebras