math lessons - Karoubi envelope

The Karoubi envelope is a classification of the idempotents of a category. Precisely, given a category C, an idempotent of C is an endomorphism $e: A \rightarrow A$ with e² = e.

The Karoubi envelope of C, sometimes written Split(C), is a category with objects pairs of the form (A, e) where $e : A \rightarrow A$ is an idempotent of C, and morphisms triples of the form

$(e, f, e^{\prime}): (A, e) \rightarrow (A^{\prime}, e^{\prime})$

where $f: A \rightarrow A^{\prime}$ is a C-morphism satisfying $e^{\prime} \circ f = f = f \circ e$ .

An automorphism in Split(C) is of the form $(e, f, e): (A, e) \rightarrow (A, e)$ , with inverse $(e, g, e): (A, e) \rightarrow (A, e)$ satisfying:

$g \circ f = e = f \circ g$

$g \circ f \circ g = g$

$f \circ g \circ f = f$

If the first equation is relaxed to just have $g \circ f = f \circ g$ , then f is a partial automorphism (with inverse g). A (partial) involution in Split(C) is a self-inverse (partial) automorphism.

Examples

If C has products, then given an isomorphism $f: A \rightarrow B$ the mapping $f \times f^{-1}: A \times B \rightarrow B \times A$ , composed with the "symmetric" map $sym: B \times A \rightarrow A \times B$ , is a partial involution.

Categories: Category theory

01-04-2007 01:18:14
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