By Volkodavov V.F., Radionova I.N., Bushkov S.V.

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14. 17 In a category with finite biproducts, we can add morphisms f, g : X ⇒ Y by defining f + g as the composite X ∆ G X ⊕X f ⊕g G Y ⊕Y ∇ G Y. With the zero morphism 0 : X → Y as unit, this makes every homset into a commutative monoid. The following theorem shows that this works when we start with any V-enriched category with finite products (instead of a Set-category), and moreover that this ‘lifting of enrichment’ is functorial. 18 Theorem There is a functor ( ) : BP(V-Cat) −→ (cMon(V))-Cat.

In fact, ( )∗ is an equivalence since ( )∗∗ ∼ = Id. 10. 11 Proposition Let C be a symmetric monoidal category with finite biproducts, and denote its scalars by R = C(I, I). 10 is strong monoidal. If X is a compact object in C, then C(I, X) is finitely projective. 7 shows that the compact objects in ModR are precisely the finitely projective R-modules. 20) are isomorphisms. e. 18. Hence the requirement in the previous proposition means that the tensor product of C “behaves bilinearly”. This is quite a natural restriction.

Indeed, there are evident injections κi : Xi → X defined by (κi (x))i = x and (κi (x))j = 0 for i = j. There are also evident projections πi : X → Xi determined by πi ((xj )j ) = 27 Chapter 2. Tensors and biproducts xi . 5). However, X does not satisfy the universal requirements for a (bi)product, as witnessed by the previous lemma. It is, however, universal in a restricted sense, as follows. Let us call a cone gi : Y → Xi bounded when i∈I gi 2 < ∞. Then the tuple g : Y → X is well-defined by g(y) = (gi (y))i∈I and is the unique morphism satisfying πi ◦ g = gi .