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Enriched Categories

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 compact-open 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 Davis-Januszkiewicz 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 ] .

  • enriched bicategory

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 ] .

References

[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. Cauchy-completion 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. Springer-Verlag, 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/S0022-4049(01)00048-2 .

[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, Stanley-Reisner algebras, and loop spaces. Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), 261-291, 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. 1-3, 265–283 [MR0700252], with new commentary by the author.

[Wal82]     R. F. C. Walters. Sheaves on sites as Cauchy-complete categories. J. Pure Appl. Algebra , 24(1):95–102, 1982, http://dx.doi.org/10.1016/0022-4049(82)90061-5 .