imported>Эарендил: of course Twinkle does not check this

2026-03-14T10:22:01Z

of course Twinkle does not check this

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==synonym==
"In mathematics, an autonomous category is another term for a symmetric monoidal closed category."

This is not current usage. An autonomous category is a monoidal category in which every object has a left and a right dual. A left autonomous category is a monoidal category in which every object has a left dual, etc.

Autonomous category is synonymous with compact category or rigid category. Some people use compact to mean ''symmetric'' autonomous category.

: Citation? I have only seen compact category as a synonym of compact closed category, which is not a synonym of autonomous category. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/142.68.223.218|142.68.223.218]] ([[User talk:142.68.223.218|talk]]) 23:52, 14 May 2010 (UTC) 

:: A compact closed category is a symmetric autonomous category. The dual of A is given by A "internal hom" I. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/133.5.165.4|133.5.165.4]] ([[User talk:133.5.165.4|talk]]) 05:13, 20 July 2011 (UTC) 

An autonomous category is a closed category. The internal hom of A and B is <math>B \otimes A^*</math>.

The principal example of an autonomous category is the category of vector spaces (over k) with A* given by the dual of A, Hom(A,k).

: Are you sure you are not confusing with *-autonomous categories ?

:: Yes, I am sure that I am not. But, indeed, I made a mistake in the line above. What I meant to say is that the principal example of an autonomous category is the category of ''finite-dimensional'' vector spaces. For *-autonomous categories the principal example is topological vector spaces, possibly infinite dimensional. The dualising object is of course given by the underlying field ''k''.

::[[User:Lkajdf|Lkajdf]] 12:20, 16 June 2007 (UTC)

I've added in a note about the connection between autonomous and *-autonomous categories, this relies on a (published, of course) theorem of Cockett and Seely (namely, that *-autonomous categories are the same thing as linearly distributive categories with negation), which makes clear the connection. However, this characterization of *-aut cats might not be well known and my wiki-fu is not strong enough (nor my time free enough) to properly write up the reference, together with the separate page on LDC's and LDC's with negations that is probably called for. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/137.111.240.148|137.111.240.148]] ([[User talk:137.111.240.148|talk]]) 22:47, 7 December 2008 (UTC) 

== Proposed merge of [[Autonomous category]] into [[Rigid category]] ==

Seems to be redundant. [[User:1234qwer1234qwer4|1234qwer]][[User talk:1234qwer1234qwer4|1234qwer]][[Special:Contribs/1234qwer1234qwer4|4]] 10:21, 14 March 2026 (UTC)

Talk:Autonomous category - Revision history

imported>Эарендил: of course Twinkle does not check this