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.