Examples of Enriched Categories

Here we collect examples of enriched categories .

According to the preface of Street’s paper [ ? ] , the study of additive categories was more active than that of general category theory at the beginning of category theory. In other words, enriched categories are central objects of study from the beginning of category theory.

• additive category
• linear category over a commutative ring

Categories enriched over commutative monoids are sometimes called semi-additive categories. Categories enriched over groupoids are called track categories by Baues and Pirashvili [ BP ] .

A k -linear category can be regarded as a “ k -algebra with several objects”. It can be extended to “differential graded algebra with several objects”, which is usually called a dg category (differential graded category). Categories enriched over spectra are also getting popular.

• dg category
• spectral category

Enriched categories are also used to define (strict) higher categories .

• strict n -category
• Gray-category

An exotic example of enriched category was found by Lawvere [ Law73 ] . The set of nonnegative real numbers ℝ _{≥ 0} can be made into a symmetric monoidal category by regarding it as a poset , hence a small category, by the standard total ordering and the addition. Lawvere found that any metric space can be regarded as a smal category enriched over this monoidal category.

This viewpoint allows us to apply the Euler characteristic of small categories. See papers [ LW ,  Wil ,  Lei ] by Leinster and Willerton. It is called the magnitude of a metric space.

• magnitude

In homotopy theory, especially when we use model categories , it is convenient to use categories enriched over the category of simplicial sets . More generally, we may define model categories enriched over a monoidal model category .

• simplicial category
• enriched model category

References