Weighted Limits and Colimits

Assuming only a very rudimentary knowledge of enriched category theory — V-categories, V-functors, and V-natural transformations for a closed, symmetric monoidal, complete and cocomplete category V — we introduce weighted limits and colimits, the appropriate sort of limits for the enriched setting. This “modern” approach was introduced to the author through talks by Mike Shulman at the Category Theory Seminar at the University of Chicago in Fall 2008. These notes were written in an attempt to understand and internalize the many wonderful things that he said. 1. Homs and Tensor Products of V-functors A one object category enriched in Ab is a ring, which we call R. A Ab-functor from R to Ab is a left R-module if it is covariant and a right R-module if it is contravariant. Let M : R → Ab and N : R → Ab be two such functors, and let M and N also denote the respective objects of Ab in their image. A slight modification of the usual functor tensor product to account for the fact that R and Ab are Ab-categories yields the following coequalizer in Ab: M ⊗Z R⊗Z N (m,r,n)7→(m,rn) // (m,r,n) 7→(mr,n) // M ⊗Z N // M ⊗R N. This constructs the tensor product over R of a right R-module with a left R-module using the monoidal structure (Ab,⊗Z,Z). More generally, let (V,⊗, I) be a closed, symmetric monoidal category that is complete and cocomplete. Given a V-category C and V-functors F : C → V and G : C → V a generalization of the above construction yields the V-tensor product of F and G, an object F ⊗C G of V: ∐ a,b∈C Fb⊗ C(a, b)⊗Ga // // ∐ c∈C Fc⊗Gc // F ⊗C G. 1 The top map of the parallel pair is induced by the composite Fb⊗ C(a, b)⊗Ga a,b// Fb⊗ V(Fb, Fa)⊗Ga ev⊗1 // Fa⊗Ga ↪→ ∐ c Fc⊗Gc, where Fa,b is the arrow of V given because F is a V -functor and ev is the evaluation map, the counit of the adjunction on V of the monoidal product with the internalhom. The bottom map is induced by a similar composite with G in place of F . The dual notion gives an enriched hom of V-functors G : C→ D and H : C→ D between V-categories C and D (note that the codomain category need no longer Date: July 9, 2009. 1The C(a, b) that appears on the left is the internal-hom, an object of V. This is the default; hom-sets will explicitly noted as such.