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Stream: theory: category theory

Topic: Categories with hom but no tensor

fosco (Jun 30 2020 at 09:10):

We are used to consider monoidal categories, i.e. $\mathcal C$ that are internal pseudomonoids $\otimes : \mathcal C \times \mathcal C \to \mathcal C$.

But what about considering categories endowed with a bifunctor $[-,-] : \mathcal C^\text{op}\times \mathcal C \to \mathcal C$, where no tensor is required, and maybe $[-,-]$ enjoys some properties? Has anyone done this?

Paolo Capriotti (Jun 30 2020 at 09:13):

They're called closed categories.

fosco (Jun 30 2020 at 09:24):

Ah, nice! I called closed categories those that also had a monoidal structure, which is moreover closed

Mike Shulman (Jul 01 2020 at 16:07):

Those are usually called "closed monoidal categories" (or perhaps "monoidal closed categories" since one can take either the monoidal or the closed structure as basic and characterize the other in terms of it by a universal property).

Mike Shulman (Jul 01 2020 at 16:08):

Note that the traditional definition of "closed category" has a unit object $I$ in addition to a bifunctor $[-,-]$, but there is a more recent variant called "prounital closed category" that has only $[-,-]$.