Enriched category theory
Enriched category theory , not to be confused with internal category theory, generalizes category theory by replacing hom-sets with hom-objects in a general symmetric monoidal category. The theory works best when the base category is also complete, cocomplete, and closed, in which case it is called a cosmos .
Examples and applications
V | sets | booleans \(\{\top,\bot\}\) | \([0,\infty]\) | weighted sets |
V-Cat | categories | preordered sets | Lawvere metric spaces | normed categories |
V-2-Cat | 2-categories | locally preordered 2-categories | metric categories |
Note: The concept I call a “metric category” seems not to have a standard name.
Metrics and norms
References on Lawvere metric spaces, normed categories, and related topics are in the analysis section. See also:
- Hofmann, Seal, Tholen, eds., 2014: Monoidal topology: A categorical approach to order, metric, and topology (doi, pdf)
Tight and loose morphisms
Enriched category theory can be used to describe categories with two kinds of morphisms, a strict kind called tight and a less strict kind called loose. The nLab calls these M-categories (or, starting from a different motivation, relative categories ) and their 2-categorical generalization F-categories .
- Power, 2002: Premonoidal categories as categories with algebraic structure
(doi)
- Mentions M-categories as Subset-categories in Example 2
- Lack & Shulman, 2012: Enhanced 2-categories and limits for lax morphisms (doi,
arxiv, nCat Cafe )
- Introduces F-categories, which are 2-categories having tight and loose 1-morphisms with compatible 2-morphisms
- Lack’s slides about F-categories from CT 2008
- Ahrens et al, 2021: The univalence principle (arxiv), Example 12.2: M-categories & Example 12.3: F-bicategories
Enriched categories
Books and notes
- Kelly, 1982: Basic concepts of enriched category theory (pdf)
- The standard reference, famously unfriendly
- Borceux, 1994: Handbook of Categorical Algebra, Vol. 2, Ch. 6: Enriched category theory
- Riehl, 2014, Categorical Homotopy Theory, Ch. 3: Basic concepts of enriched
category theory & Ch. 7: Weighted limits and colimits
- Fairly readable account of the general theory, a good place to start
- May, notes : Symmetric monoidal categories and enriched categories (pdf)
General theory
- Cruttwell, 2014, PhD thesis: Normed spaces and change of base for enriched
categories (pdf)
- Comprehensive treatment of base changes between enriched categories
- Sec. 4.4: How adjunctions between base categories V and W extend to adjunctions between V-Cat and W-Cat
- Vasilakopoulou, 2019: Enriched duality in double categories: V-categories
and V-cocategories (doi, arxiv)
- Extensively uses V-Mat, the double category of matrices with values in V
Limits and colimits
- Riehl, 2009, lecture notes: Weighted limits and colimits (pdf)
- Based on Shulman’s 2008 lectures at U. Chicago’s Category Theory Seminar
- Wolff, 1974: V-cat and V-graph (doi)
- Lack & Rosicky, 2012: Enriched weakness (doi, arxiv)
- Weak limits, weak adjoints, and injectivity in enriched categories
Other topics
Enriched monoidal categories
One can also consider monoidal categories and SMCs enriched in another SMC. While apparently well-known to experts, it is difficult to find literature on this topic.
- Morrison, Penneys, 2017: Monoidal categories enriched in braided monoidal
categories (doi, arxiv, slides)
- While mostly about generalization to the braided case, a clear definition of a strict V-monoidal category is given in Section 2
Linear monoidal categories
An important special case is linear monoidal categories, or monoidal categories enriched in Vect. Linear (symmetric) monoidal categories are one possible categorification of a (commutative) ring.
- Savage, 2019: String diagrams and categorification (arxiv)
- Liu, 2018: Presentations of linear monoidal categories and their endomorphism algebras (arxiv)
- Baez, Moeller, Trimble, 2021: Schur functors and categorified plethysm (arxiv)
- This paper takes a 2-rig to be linear symmetric monoidal category that is Cauchy complete (Definition 1.1)
- General results about 2-rigs scattered throughout, such as Theorem 3.3 ff.