Semimodule Enrichment

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to hom-semimodules) are functorial, and use them to make precise the intuition that ''compact objects are finite-dimensional'' in standard cases.

[1]  M. Laplaza,et al.  Coherence for distributivity , 1972 .

[2]  Marcelo P. Fiore,et al.  Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic , 2007, TLCA.

[3]  G. M. Kelly,et al.  BASIC CONCEPTS OF ENRICHED CATEGORY THEORY , 2022, Elements of ∞-Category Theory.

[4]  Thomas A. O. Fox.  Coalgebras and cartesian categories , 1976 .

[5]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[6]  Samson Abramsky,et al.  Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories , 2005, CALCO.

[7]  Bart Jacobs,et al.  Semantics of Weakening and Contraction , 1994, Ann. Pure Appl. Log..

[8]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[9]  B. Mitchell,et al.  Theory of categories , 1965 .

[10]  P. Hô.  An Embedding Theorem for Abelian Monoidal Categories , 2002, Compositio Mathematica.

[11]  P. H. Hai.  An Embedding theorem for abelian monoidal categories , 2000, math/0004160.

[12]  A. Kock.  Monads on symmetric monoidal closed categories , 1970 .

[13]  B. Eckmann,et al.  Group-like structures in general categories I multiplications and comultiplications , 1962 .

[14]  G. M. Kelly,et al.  Coherence for compact closed categories , 1980 .

[15]  J. Golan.  Semirings and their applications , 1999 .

[16]  On comonadicity of the extension-of-scalars functors , 2005, math/0510272.

[17]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[18]  G. M. Kelly,et al.  Coherence in categories , 1972 .