An enriched category is a category in which the
“set of morphisms” between two objects is
an object of another category. For example, in the
category of topological spaces and continuous maps,
the set of morphisms Map(
X,Y
) between two objects
X
and
Y
can be made into an object in the category of
topological spaces by using the
compactopen topology
.
Such a structure is called a Cartesian closed
category, i.e. a category enriched over itself.
More generally, there is a notion of categories
enriched over a
monoidal category
. One of the most famous books for enriched categories
is Kelly’s book
[
Kel82
]
. Another choice is the chapter 6 in the second volume
of Borceux’s
[
Bor94
]
.

a category enriched over a monoidal category
V
(
V
category).

a
V
functor between
V
categories.

a
V
natural transformation between
V
functors.
Typical examples include additive categories and
Abelian categories
used in
homological algebra
. There are many other important examples used in
mordern homological algebra, such as
dg categories
and
spectral categories
. We collect examples in the following page.
When we deal with these categories, it’s useful
to know general properties of enriched categories. For
example, the fact that the category of dg categories
has a structure of a symmetric
monoidal category
follows from the next property.

The category of categories enriched over a symmetric
monoidal category has a structure of a symmetric
monoidal category.
When a category
C
is enriched by itself, the internal Hom functor is
deﬁned. Such a category is called a closed
monoidal category. There is a variation in which
internal Hom functor is deﬁned without assuming
a monoidal structure, i.e. closed category introduced
by Eilenberg and Kelly
[
EK66
]
.

closed monoidal category

closed category
Limits and colimits
in enriched categories are studied by Kelly’s
book
[
Kel82
]
, in which they are called indexed (co)limits. They
are called weighed (co)limits in
[
KS
]
by Kelly and Schmitt.
They are used, for example, by McClure,
Schwänzl, Vogt
[
MSV97
]
in their study of
topological Hochschild homology
and by Panov, Ray, and Vogt
[
PRV
]
in their study of the
DavisJanuszkiewicz spaces
.
Kan extensions
in enriched categories can be found in Kelly’s
book and Dubuc’s paper
[
Dub70
]
.

Kan extensions in enriched categories
The notion of monoidal categories has been
generalized in various directions
, with which enriched categories are generalized. For
example, categories enriched over a lax monoidal
category are used in
[
BW
]
by Batanin and Weber.
Bicategories
enriched over the category of symmetric monoidal
categories are studied by Guillou in
[
Gui
]
.
According to Batanin and Markl
[
BM
]
, we’d better use enrichments by duoidal
category in order to consider
monoidal structures
on enriched categories. Duoidal categories are
categories having two kinds of monoidal structures.
Since a monoidal category can be regarded as a
bicategory with a single object, we may extend
enrichments to those by
bicategories
. According to Street
[
Str05
]
, such a structure has been studied for quite a while.
For example, there are papers by Walter
[
Wal82
]
and Betti and Carboni
[
BC82
]
.

categories enriched over a bicategory
A more recent work includes
[
KLSS02
]
by Kelly, Labella, Schmitt, and Street.
We may also deﬁne bicategories enriched over
monoidal bicategories. See Hoffnung’s paper
[
Hof
]
.
A modern approach is proposed by Bacard
[
Bacb
]
. He deﬁnes a category enriched over a
bicategory in which a class morphisms is designated as
weak equivalences.
Leinster’s homotopy monoids
and Segal categories can be treated in this way.
Bacard also introduced weakly enriched categories
over a
monoidal model category
in
[
Baca
]
.
[Baca]
Hugo V. Bacard. Lax Diagrams and Enrichment,
arXiv:1206.3704
.
[Bacb]
Hugo V. Bacard. Segal Enriched Categories I,
arXiv:1009.3673
.
[BC82]
R. Betti and A. Carboni.
Cauchycompletion and the associated sheaf.
Cahiers Topologie G
éom. Diff
érentielle
, 23(3):243–256, 1982.
[BM]
M. Batanin and M. Markl. Centers and
homotopy centers in enriched monoidal categories,
arXiv:1109.4084
.
[Bor94]
Francis Borceux.
Handbook of categorical algebra. 2
, volume 51 of
Encyclopedia of Mathematics and its Applications
. Cambridge University Press, Cambridge, 1994.
[BW]
Michael Batanin and Mark Weber. Algebras of higher
operads as enriched categories,
arXiv:0803.3594
.
[Dub70]
Eduardo J. Dubuc.
Kan extensions in enriched category theory
. Lecture Notes in Mathematics, Vol. 145.
SpringerVerlag, Berlin, 1970.
[EK66]
Samuel Eilenberg and G. Max Kelly. Closed
categories. In
Proc.
Conf. Categorical Algebra (La Jolla, Calif., 1965)
, pages 421–562. Springer, New York, 1966.
[Gui]
Bertrand Guillou. Strictiﬁcation of
categories weakly enriched in symmetric monoidal
categories,
arXiv:0909.5270
.
[Hof]
Alexander E. Hoffnung. The Hecke Bicategory,
arXiv:1007.1931
.
[Kel82]
Gregory Maxwell Kelly.
Basic concepts of enriched category
theory
, volume 64 of
London Mathematical Society Lecture Note
Series
. Cambridge University Press, Cambridge, 1982.
[KLSS02]
Max Kelly, Anna Labella, Vincent Schmitt, and Ross
Street. Categories enriched on two sides.
J. Pure Appl. Algebra
, 168(1):53–98, 2002,
http://dx.doi.org/10.1016/S00224049(01)000482
.
[KS]
G.M. Kelly and V. Schmitt. Notes on enriched
categories with colimits of some class (completed
version),
arXiv:math.CT/0509102
.
[MSV97]
J. McClure, R. Schwänzl, and
R. Vogt.
THH
(
R
)
R
⊗
S
^{
1
}
for
E
_{
∞
}
ring spectra.
J. Pure Appl. Algebra
, 121(2):137–159, 1997.
[PRV]
Taras Panov, Nigel Ray, , and Rainer Vogt. Colimits,
StanleyReisner algebras, and loop spaces.
Categorical decomposition techniques in algebraic
topology (Isle of Skye, 2001), 261291, Progress in
Mathematics, 215, Birkhäuser, Basel, 2004,
arXiv:math.AT/0202081
.
[Str05]
Ross Street. Enriched categories and cohomology.
Repr. Theory Appl. Categ.
, (14):1–18, 2005,
http://www.tac.mta.ca/tac/reprints/articles/14/tr14abs.html
. Reprinted from Quaestiones Math.
6
(1983), no. 13, 265–283 [MR0700252], with new
commentary by the author.
[Wal82]
R. F. C. Walters. Sheaves on sites as
Cauchycomplete categories.
J. Pure Appl. Algebra
, 24(1):95–102, 1982,
http://dx.doi.org/10.1016/00224049(82)900615
.