Magmal characterisations of cocartesian categories

Abstract

We present a survey of characterisations of cocartesian categories in terms of monoidal categories - and, more generally, magmal categories - satisfying additional properties. In particular, we show that the following are equivalent for a unital magmal category $(\mathcal{M},\otimes )$, sharpening several classical characterisations. * $(\mathcal{M},\otimes )$ is cocartesian monoidal. * Every object of $\mathcal{M}$ admits the structure of a unital magma with respect to $\otimes$, such that every morphism is a homomorphism, and a single compatibility condition holds between the magma structures and $\otimes$. * The tensor product functor $\otimes :\mathcal{M}\times \mathcal{M}\to \mathcal{M}$ admits a right adjoint.

13 pages