In 2016, Garner and Schappi gave a criterion for existence of finite biproducts in specific class monoidal categories. We provide an elementary proof (a slight generalization of) their result. Also, we explain how to prove, by using the same technique, analogous result including infinite biproducts.

Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the global sections a sheaf local categories on topological space. stiff embeds into such sections. An infinitary version these theorems also holds in spatial case. These representation results are functorial and subsume Lambek–Moerdijk–Awodey for toposes, Stone Boolean algebras, Takahas...

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed...

At the heart of the axiomatic formulation of 1+1-dimensional topological field theory is the set of all surfaces with boundary assembled into a category. This category of surfaces has compact 1-manifolds as objects and smooth oriented cobordisms as morphisms. Taking disjoint unions gives a monoidal structure and a 1+1-dimensional topological field theory can be defined to be a monoidal functor ...