Talk:Autonomous category
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synonym
[edit]"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 unsigned comment added by 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 unsigned comment added by 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 .
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.
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 unsigned comment added by 137.111.240.148 (talk) 22:47, 7 December 2008 (UTC)