# Algebra, Topology, and Category Theory

## 1st Edition

### A Collection of Papers in Honor of Samuel Eilenberg

**Editors:**Alex Heller Myles Tierney

**eBook ISBN:**9781483262611

**Imprint:**Academic Press

**Published Date:**28th January 1976

**Page Count:**238

## Description

Algebra, Topology, and Category Theory: A Collection of Papers in Honor of Samuel Eilenberg is a collection of papers dealing with algebra, topology, and category theory in honor of Samuel Eilenberg. Topics covered range from large modules over artin algebras to two-dimensional Poincaré duality groups, along with the homology of certain H-spaces as group ring objects. Variable quantities and variable structures in topoi are also discussed.

Comprised of 16 chapters, this book begins by looking at the relationship between the representation theories of finitely generated and large (not finitely generated) modules over an artin algebra. The reader is then introduced to reduced bar constructions on deRham complexes; some properties of two-dimensional Poincaré duality groups; and properties invariant within equivalence types of categories. Subsequent chapters explore the work of Samuel Eilenberg in topology; local complexity of finite semigroups; global dimension of ore extensions; and the spectrum of a ringed topos.

This monograph will be a useful resource for students and practitioners of algebra and mathematics.

## Table of Contents

List of Contributors

Preface

Large Modules over Artin Algebras

Introduction

1. Preliminary Results

2. Proof of Theorem A

3. Proof of Theorem B

References

Reduced Bar Constructions on deRham Complexes

1. Bar Construction on DG Algebras

2. The Eilenberg-Moore Spectral Sequence

3. A Pairing of B(A) and a Cobar Construction

4. The E1 Term of the Cobar Construction

5. Applications

References

Flatness and ProjectIvity of Modules That Come from ℂ-Sets

1. Nonadditive Flatness

2. Flatness of ℤM

3. Nonadditive Projectivity

4. Projectivity of ℤM

References

Some Properties of Two-Dimensional Poincaré Duality Groups

1. Preparatory Material

2. Proof of Theorem A

3. Proof of Theorem B

4. The Pro-p-Completion of a Surface Group

References

Properties Invariant within Equivalence Types of Categories

1. The Diagrammatic Language

2. The Theorem

3. Back to Frege

Coherence for the Tensor Product of 2-Categories, and Braid Groups

Introduction

1. The Tensor Product

2. The N-Dimensional Cube QN as a 2-Category

3. Proofs of the Lemmata

References

Homology of Certain H-Spaces as Group Ring Objects

1. Ring Objects

2. Additive and Multiplicative Group Functors

3. Existence of the Group Ring Functor

4. Basic Classes of Examples of Group Rings

5. The Group Ring H

*+(Z x FΨq, k)*

6. The Group Rings H(Z x BGL(Fq), k) and H*(Z x BGL(Fq), k)

6. The Group Rings H

References

A Whitehead Theorem

1. Introduction

2. The Bousfield Factorization

3. The Whitehead Theorem

References

Variable Quantities and Variable Structures in Topoi

1.

2.

3.

References

The Work of Samuel Eilenberg in Topology

Bibliography

What It Means for a Coalgebra To Be Simply Connected

References

Local Complexity of Finite Semigroups

1. Complexity Axioms

2. Elementary and Type V Subsemigroups

3. The Largest Local Complexity Function

References

The Global Dimensions of Ore Extensions and Weyl Algebras

Introduction

1. Ore Extensions

2. Weyl Algebras

3. Examples

References

Global Dimension of Ore Extensions

1. Introduction

2. Tor Functor and Intersections

3. Ore Extensions

4. The Commutative Case

References

On the Spectrum of a Ringed Topos

1. Localization, Primes, Local Rings, Etc.

2. The Problem of the Spectrum and Some Previous Solutions

3. A New Method

References

Forcing Topologies and Classifying Topoi

1. Forcing Topologies

2. The Classifying Topos

References

Published Works of Samuel Eilenberg

## Details

- No. of pages:
- 238

- Language:
- English

- Copyright:
- © Academic Press 1976

- Published:
- 28th January 1976

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483262611