Left Adjoint: Free
Forgetful functors tend to have left adjoints, which are 'free' constructions. For example:
- free module: the forgetful functor from (the category of -module) to has left adjoint, with, the free -module with basis .
- free group
- free lattice
- tensor algebra
- free category, adjoint to the forgetful functor from categories to quivers
For a more extensive list, see (Mac Lane 1997).
As this is a fundamental example of adjoints, we spell it out: adjointness means that given a set X and an object (say, an R-module) M, maps of sets correspond to maps of modules : every map of sets yields a map of modules, and every map of modules comes from a map of sets.
In the case of vector spaces, this is summarized as: "A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."
Symbolically:
The counit of the free-forget adjunction is the "inclusion of a basis": .
Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.
Read more about this topic: Forgetful Functor
Famous quotes containing the words left and/or free:
“Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, they have their reward. But when thou doest alms, let not thy left hand know what thy right hand doeth.”
—Bible: New Testament Jesus, in Matthew, 6:2-3.
From the Sermon on the Mount.
“Most men, even in this comparatively free country, through mere ignorance and mistake, are so occupied with the factitious cares and superfluously coarse labors of life that its finer fruits cannot be plucked by them. Their fingers, from excessive toil, are too clumsy and tremble too much for that.”
—Henry David Thoreau (18171862)