nLab weak adjoint

Idea

A weak adjoint is like an adjoint functor but without the uniqueness of factorizations.

Importantly, this means that unlike adjoint functors, weak adjoints are not unique, and so a weak adjoint of a given functor is a structure whereas a functor having an adjoint is a property.

Definition

Let $G:D\to C$ be a functor. We say $G$ has a left weak adjoint if for each $x\in C$ there exists a morphism $\eta_x:x\to G z$ such that for any morphism $f:x\to G y$ there exists a (not necessarily unique) morphism $g:y\to z$ such that $f = G g \circ\eta_x$.

Examples

• A weak limit is a weak right adjoint to a constant-diagram functor.

Remarks

Of course, if the factorizations $g$ are always unique, this is precisely an ordinary left adjoint. A weak adjoint can also be regarded as a stronger form of the solution set condition in which the solution sets are required to be singletons. The “dual” property, in which the solution sets need not be singletons but the factorizations are unique, is called a multi-adjoint.

Related Concepts

• weak representation of a functor

• weakly initial object

References

Weak adjoint functors along with weak colimits were defined in:

• Paul Kainen, Weak adjoint functors, Mathematische Zeitschrift 122 1 (1971) 1-9 [dml:171575]