*-autonomous categories

A *-autonomous category is a symmetric monoidal closed category such that ...

Please expand, including both symmetric and non-symmetric versions, and short definition due to Ehrhard.

References

An short discussion of *-autonomous categories appears in sections 4.8, 4.13, the survey

Melliès, Paul-André. 2009. ‘Categorical Semantics of Linear Logic’. In Panoramas et Synthèses 27: Interactive Models of Computation and Program Behaviour, edited by Pierre-Louis Curien, Hugo Herbelin, Jean-Louis Krivine, and Paul-André Melliès. Société Mathématique de France. pdf

@incollection{mellies_categorical_2009,
	title = {Categorical {Semantics} of {Linear} {Logic}},
	isbn = {978-2-85629-273-0},
	url = {http://www.pps.univ-paris-diderot.fr/~mellies/papers/panorama.pdf},
	booktitle = {Panoramas et synthèses 27: {Interactive} models of computation and program behaviour},
	publisher = {Soci\'{e}t\'{e} Mathématique de France},
	author = {Melliès, Paul-André},
	editor = {Curien, Pierre-Louis and Herbelin, Hugo and Krivine, Jean-Louis and Melli\`{e}s, Paul-Andr\'{e}},
	year = {2009}
}

*-autonomous categories were first defined and studied by Michael Barr.

Barr, Michael. 1991. ‘*-Autonomous Categories and Linear Logic’. Mathematical Structures in Computer Science 1 (2): 159–78. https://doi.org/10.1017/S0960129500001274.

@article{barr_1991,
	title = {*-{Autonomous} categories and linear logic},
	volume = {1},
	issn = {14698072},
	doi = {10.1017/S0960129500001274},
	number = {2},
	journal = {Mathematical Structures in Computer Science},
	author = {Barr, Michael},
	year = {1991},
	pages = {159--178}

Barr, Michael. 1995. ‘Nonsymmetric ∗-Autonomous Categories’. Theoretical Computer Science 139: 115–30. https://doi.org/10.1016/0304-3975(94)00089-2.

@article{barr_1995,
	title = {Nonsymmetric ∗-autonomous categories},
	volume = {139},
	doi = {10.1016/0304-3975(94)00089-2},
	language = {en},
	journal = {Theoretical Computer Science},
	author = {Barr, Michael},
	year = {1995},
	pages = {115--130}
}