# Groups and Operator Algebras Seminar

**Speaker:** Kevin Piterman (Philipps-Universität Marburg)

**Title:** A categorical approach to study posets of decompositions into subobjects.

**Abstract:** Given a sequence of groups G_n with inclusions G_n -> G_{n+1}, an important question in group (co)homology is whether there is homological stability. That is, if for a given integer j, there is some m such that for all n>m, the map H_j(G_n) -> H_j(G_{n+1}) is an isomorphism. To detect this behaviour one usually constructs a family of highly connected simplicial complexes K_n on which the groups G_n naturally act. For example, for the linear groups GL_n or SL_n, K_n can be the Tits building or the complex of unimodular sequences, while for the automorphism group of the free groups F_n one can take the complex of free factors.

In this talk, we discuss a categorical framework that describes these constructions in a unified way. More precisely, for an initial symmetric monoidal category C, we take an object X and consider the poset of subobjects of X. From this bounded poset, we take only those subobjects which are complemented, i.e. x \vee y = 1 and x \wedge y = 0, and the join operation coincides with the monoidal product. The monoidal product should be interpreted as the "expected" coproduct of the category. Thus, for the free product in the category of groups, if we start with a free group of finite rank then the complemented subobject poset is exactly the poset of free factors, and for the category of vector spaces with the direct sum we obtain the subspace poset. From this construction, we define related combinatorial structures, such as the poset of (partial) decompositions or the complex of partial bases, and establish general properties and connections among these posets. Finally, we specialise these constructions to matroids, modules over rings, and vector spaces with non-degenerate forms, where there are still many open questions.

**GOA website:** https://sites.google.com/view/copenhagen-goa-seminar