Cohomological results in monoid and category theory via classifying spaces

Abstract

The thesis lays foundations for a topological, homotopical, and homological methodology in the study of monoids via the classifying space functor ||: CAT → TOP. In Section 0 basic constructions are described giving relationships between categories, monoid actions, and posets. An important adjunction D: CAT ↓ C ⇄ CAT C op:∗ is described that will be used extensively throughout the work. Then the classifying space functor is introduced and several known homotopy theoretical results are stated. Using these results, it is shown that when the classifying space functor is applied to the adjunction above, one gets maps that are homotopy equivalences. This has far-reaching implications. The (co)homology of the classifying spaces is then described and it is proved that this (co)homology arises from the comonad associated with the above adjunction. This is a direct generalization of the fact that the bar construction arises from an adjunction. Let X be a right M-set with M a given monoid. Then it is proved that H ∗(M, ZX) is isomormphic to the classifying space of a certain category naturally associated with the ( M, X). The section ends with conditions under which the classifying spaces of a monoid is homotopy equivalent to a finite-dimensional CW-complex. In Section 1, it is proved that associated to every functor F: X → Y is a spectral sequence whose E p, q 2 term is H p ( Y op , H q DF) and whose termination is H p + q ( X, Z). Here, DF: Y op → CAT is the functor associated with F where D is as in the above adjunction. Thus, H q DF = H q ( DF(), Z) → AB as a functor. One should anticipate this result, since |X| →|DF ∗ Y | from Section 0, and DF ∗ Y → Y should be thought of as a categorical analogue of a fiber bundle with “fiber” DF. Then, using the results of Section 0, we obtain that given f: C op → CAT, there exists a spectral sequence whose E p, q 2 term is H q(C op, H qƒ) and whose termination is H p + q(|ƒ∗C|, Z) . Here, H qƒ:C op → Ab is given by H qƒ() = H q(|ƒ(·)|, Z) . The first spectral sequence is applied to the following situation to compute the Euler characteristic of | C| when it exists: F: C → P is a functor from C to a finite poset such that H q DF = 0 if q is large enough, and finitely generated otherwise. The section ends with conditions under which the right M-module Z where i · m = 0 if m ≠ 1 (we assume here that the group of units of M = {1}) is acyclic. This module is of importance in the next section. The last section deals with the issue of when a surmorphism is a (co)homological equivalence. It is shown that ƒ:M → N is a homology equivalence iff |Dƒ| is acyclic. Then, it is proved that if ƒ:M → N is given and I ⊆ M is an ideal such that ƒ¦M − I is injective as a function, then if f∣ I is a homology equivalence, so if ƒ. This result, proved using the spectral sequence of Section 1, should be compared to the following result in homotopy theory: Let ( X, A) be an NDR pair and suppose ƒ:A → B is a homotopy equivalence. Then so is \\ ̂ tf:X → X ∪ ƒB . Then null surmorphisms are defined and it is proved that these surmorphisms are (co)homology equivalences.