Magmal characterisations of cocartesian categories
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912539354333184 |
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| author | Arkor, Nathanael |
| author_facet | Arkor, Nathanael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Magmal characterisations of cocartesian categories Arkor, Nathanael Category Theory 18A35, 18A40, 18M05 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. |
| title | Magmal characterisations of cocartesian categories |
| topic | Category Theory 18A35, 18A40, 18M05 |
| url | https://arxiv.org/abs/2508.11615 |