Enriched category theory

Table 1: My favorite examples of iteratively enriched categories

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

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)
  • Proves that if V is cocomplete, then so are V-Graph (Prop 2.4) and V-Cat (Cor. 2.14)
  • Relevant MO questions (1 , 2 )
• Lack & Rosicky, 2012: Enriched weakness (doi, arxiv)
  • Weak limits, weak adjoints, and injectivity in enriched categories

Other topics

• Clementine & Montoli, 2020: On the categorical behaviour of V-groups (arxiv)
  • Studies V-groups, which are monoid objects in V-Cat that form groups
  • Here V is a commutative, unital quantale

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)
  • Defines a 2-rig to be a linear symmetric monoidal category that is Cauchy complete (Definition 1.1)
  • General results about 2-rigs scattered throughout, such as Theorem 3.3 ff.