Magmal characterisations of cocartesian categories

TL;DR

This paper surveys various characterisations of cocartesian categories through monoidal and magmal categories, establishing equivalences and sharpening classical results in category theory.

Contribution

It provides new equivalences for unital magmal categories, linking cocartesian monoidal structures with magma structures and adjoint functors.

Findings

Cocartesian monoidal categories characterized by magma structures.

Equivalence between cocartesian properties and the existence of right adjoints.

Sharpened classical 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 ${\otimes} \colon \mathcal M \times \mathcal M \to \mathcal M$ admits a right adjoint.