Representation Theory of Combinatorial Categories

ABSTRACT......Page 1
Introduction......Page 10
This thesis in broad strokes......Page 12
Outline......Page 13
What is a representation?......Page 14
Categories and their representations......Page 15
Matrices and additive categories......Page 16
Matrices as presentations......Page 18
A familiar example of a finite presentation over a category......Page 19
Another beginning example......Page 20
Configurations of distinct points in C......Page 22
Monomials with combinatorial indexing......Page 23
When D is a finite group?......Page 25
When D is a graded polynomial algebra?......Page 27
The dimension of a category D......Page 29
Representation stability......Page 30
Basic definitions of category theory......Page 32
R-linear categories......Page 33
Representations and the category ModRD......Page 34
Matrices over a category with coefficients in a ring......Page 36
Yoneda's lemma and basic projectives in the category ModRD......Page 37
Finitely generated representations of a category......Page 39
Idempotents and projectives......Page 40
Left and right Kan extensions......Page 41
Construction of the functors f! and f*......Page 44
Two easy facts about the functors f!, f*, and f*......Page 46
Simple representations, finite length representations......Page 48
The Krull-Schmidt decomposition......Page 49
Length and the three functors f!, f*, and f*......Page 50
Projective covers and injective hulls......Page 52
The intermediate extension functor f!*......Page 53
Filtered categories, associated graded categories, and the classification of irreducible representations......Page 55
A corollary in the representation theory of algebras......Page 56
Computing the dimensions of the irreducible representation i!*p*W......Page 57
The object preorders d......Page 58
Useful facts about the preorders d......Page 60
Upper bounds for the preorder d......Page 64
Dimension zero categories......Page 68
Computing simple multiplicities from a presentation matrix......Page 70
Bounded models......Page 72
The Dold-Kan Correspondence......Page 74
Further examples of categories of dimension zero......Page 75
Schur projectives......Page 76
D-regular functions......Page 78
Summary of results for the representation theory of the category of finite sets......Page 82
Computing in D......Page 84
Computing the preorders d......Page 85
A first pass at the irreducible representations......Page 88
A Morita equivalent category separating columns and non-columns......Page 92
Computing the preorders  for the Morita equivalent category C......Page 93
Explicit construction of the irreducibles......Page 94
What we know after constructing the irreducibles......Page 97
Experimentally computing the indecomposable injectives......Page 98
Construction of the indecomposable injectives......Page 101
The multiplicity theorem for presentations......Page 103
Demonstration computation......Page 104
A computer program......Page 105
BIBLIOGRAPHY......Page 106