# A Category of Concrete Monoids

## Author

**Andrew Solomon**

## Status

Research Report 96-7

Date: February 1996, revised April 1996

## Abstract

A concrete monoid over a category C is a subset of the endomorphisms of
an object of C, containing the identity and closed under
composition. To contrast, an abstract monoid is just a one object category.
There is a natural notion of morphism between concrete monoids distinct from
the usual morphism of abstract monoids. This type of morphism is identified via
an example, and then defined, giving rise to the category of concrete monoids over C.

The utility of these definitions is explored via applications to the
Theories of Semigroups, Matrices, Vines and Automata.
With these definitions, it is possible for the first time
to make the action monoid construction into a functor
whose domain is the usual category of automata.

## Key phrases

monoid. category. concrete. semigroup. automata. vine. matrices. regular category.
bicategory. relation. division. filter.

## Content

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