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adjoint functor
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(Definition)
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Let
and
be (small) categories, and let
and
be covariant functors. is said to be a left adjoint functor to (equivalently, is a right adjoint functor to ) if there is a natural equivalence
Here the functor
is a bifunctor
which is contravariant in the first variable, is covariant in the second variable, and sends an object to
. The functor
is defined analogously.
This definition needs a bit of explanation. Essentially, it says that for every object in and every object in there is a function
which is a natural bijection of hom-sets. Naturality means that if
is a morphism in
and
is a morphism in
, then the diagram
is a commutative diagram. If we pick any
, then we have the equation
If
is a left adjoint of
, then we say that the ordered pair is an adjoint pair, and the ordered triple an adjunction from
to
, written
where is the natural equivalence defined above.
An adjoint to a functor is in some ways like an inverse (as in the case of an adjoint matrix); often formal properties about a functor lead to formal properties of its adjoint (for example the right adjoint to a left-exact functor takes injectives to injectives). An adjoint to any functor is unique up to natural isomorphism.
Examples:
- Let
be a commutative ring, and fix an -module . Let
be the functor
and let
given by
Then one can show that
is the left adjoint to
. This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.
- Let
be the forgetful functor (i.e. takes topological spaces to their underlying sets, and continuous maps to set functions). Then is right adjoint to the functor
which gives each set the discrete topology.
- If
is again the forgetful functor, this time on the category of groups, the functor
which takes a set to the free group generated by is left adjoint to .
- 1
- Daniel M. Kan. Adjoint functors. Transactions of the American Mathematical Society, Vol. 87, No. 2, (1958), 294-329.
- 2
- S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer-Verlag, 1997.
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"adjoint functor" is owned by mps. [ full author list (6) | owner history (5) ]
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Cross-references: generated by, free group, groups, discrete topology, set functions, continuous maps, topological spaces, forgetful functor, fix, commutative ring, natural isomorphism, adjoint matrix, inverse, ordered pair, equation, commutative diagram, morphism, bijection, function, object, variable, bifunctor, natural equivalence, covariant functors, categories
There are 25 references to this entry.
This is version 20 of adjoint functor, born on 2002-02-25, modified 2007-07-29.
Object id is 2691, canonical name is AdjointFunctor.
Accessed 12495 times total.
Classification:
| AMS MSC: | 18A40 (Category theory; homological algebra :: General theory of categories and functors :: Adjoint functors ) |
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Pending Errata and Addenda
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