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Concept
Compact closed category
Summary
In , a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the having finite-dimensional vector spaces as s and linear maps as s, with tensor product as the structure. Another example is , the category having sets as objects and relations as morphisms, with .
A is compact closed if every object has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and is denoted .
In a bit more detail, an object is called the dual of if it is equipped with two morphisms called the and the counit , satisfying the equations
and
where are the introduction of the unit on the left and right, respectively, and is the associator.
For clarity, we rewrite the above compositions diagrammatically. In order for to be compact closed, we need the following composites to equal :
and :
More generally, suppose is a , not necessarily symmetric, such as in the case of a pregroup grammar. The above notion of having a dual for each object A is replaced by that of having both a left and a right adjoint, and , with a corresponding left unit , right unit , left counit , and right counit . These must satisfy the four yanking conditions, each of which are identities:
and
That is, in the general case, a compact closed category is both left and right-, and .
Non-symmetric compact closed categories find applications in linguistics, in the area of categorial grammars and specifically in pregroup grammars, where the distinct left and right adjoints are required to capture word-order in sentences. In this context, compact closed monoidal categories are called (Lambek) pregroups.
Compact closed categories are a special case of , which in turn are a special case of .
Compact closed categories are precisely the . They are also .
Every compact closed category C admits a .
Official source
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