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.
Categories enriched over commutative monoids are
sometimes called semiadditive 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.
Enriched categories are also used to deﬁne
(strict)
higher categories
.

strict
n
category

Graycategory
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.
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
.