counit

Noun

Meaning

counit (plural counits)

English Wikipedia has an article on:Adjoint functorsWikipedia
(mathematics) In an adjunction, a natural transformation from the composition of the left adjoint functor with the right adjoint functor to the identity functor of the domain of the right adjoint functor.

Anagrams

Examples

Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit. Source: Internet

For example, one may have an algebra A with maps (the inclusion of scalars, called the unit) and a map (corresponding to trace, called the counit ). Source: Internet

Formally, one can compose the trace (the counit map) with the unit map of "inclusion of scalars " to obtain a map mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above. Source: Internet

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. Source: Internet

In practice, in bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension ( ), so in these cases the normalizing constant corresponds to dimension. Source: Internet

For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits. Source: Internet

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