Editing Talk:Autonomous category (section)

==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.   <span style="font-size: smaller;" class="autosigned">—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)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:: A compact closed category is a symmetric autonomous category. The dual of A is given by A "internal hom" I.  <span style="font-size: smaller;" class="autosigned">— 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)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

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.  <span style="font-size: smaller;" class="autosigned">—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)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->