Hodge theory: Difference between revisions

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{{Unreferenced|date=December 2009}}
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In [[mathematics]], a '''magma in a category''', or '''magma object''', can be defined in a [[Category (mathematics)|category]] with a [[cartesian product]]. This is the 'internal' form of definition of a [[binary operation]] in a category.


As '''Mag''' the [[magma category]] has [[Direct product#Categorical product|direct product]]s, the concept of an (internal) magma (or ''internal binary operation'') in '''Mag''' is defined, say
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<math>\top'\colon (X,\top)\times(X,\top)\to(X,\top)</math>
 
Since <math>\top'</math> is a [[morphism]] we must have
 
<math>(x \top' y) \top (u \top' z) = (x \top u) \top' (y \top z)</math>
 
If we want to take the original operation, this will be allowed only if the [[medial magma|medial identity]]
 
<math>(x \top y) \top (u \top z) = (x \top u) \top (y \top z)</math>
 
is valid.
This operation, which gives a [[medial magma]], can have a two-sided [[identity element|identity]] only if it is a [[commutative]] [[monoid|monoidal operation]]. The ''if'' direction is obvious.
 
As a result '''Med''', the [[medial category]], has all its objects as '' medial objects''; and this ''characterizes'' it.
 
{{DEFAULTSORT:Auto Magma Object}}
[[Category:Category theory]]

Revision as of 04:03, 25 February 2014

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