Embedding theorems for closed categories

MacDonald, Thomas (1972) Embedding theorems for closed categories. Masters thesis, Memorial University of Newfoundland.

[English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (9MB) [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

Abstract

Let us consider the following symmetric monoidal closed categories: -- (i) SM, the category of sets under the action of a commutative monoid M; in short, a category of M-sets; -- (ii) SG, the category of G-sets, where G is an abelian group; -- (iii) MK, the category of moduloids over a commutative semiring K (a moduloid is basically a monoid acted on by a semiring); -- (iv) ModK,thecategory of modules over a commutative ring K; -- (v) VF,the category of vector spaces over a field F. -- Let C be an arbitrary closed category. We are concerned with the following question: -- What conditions have to be imposed on C to ensure that it can be embedded (in some canonical way) into one or more of the above categories? -- The basic category theory needed in this thesis is provided in chapters I and II. In chapter I we have provided the details of how, in a category with biproducts, the set hom(A,B) can be given the structure of a commutative monoid (under addition). Chapter II gives a summary of the standard definitions and results leading up to the concept of a symmetric monoidal closed category. -- Since the properties of categories (i) and (iii) are not so well known, these categories are discussed in some detail in chapters III and IV. It is shown that each of the categories is in fact a symmetric monoidal closed category. -- In chapter V we answer our original question by establishing five embedding theorems. -- Each of these theorems gives sufficient conditions for a closed category to be embeddable in one of the above categories. Fairly elementary examples are given to illustrate each of the theorems. -- In the appendix a detailed example is given to show that these embeddings are not, in general, full embeddings.

Actions (login required)

View Item