An object in a category is called a retract of an object if there are morphisms and such that . In this case is called a retraction of onto .
Here may also be called a section of . (In particular if is thought of as exhibiting a bundle; the terminology originates from topology.)
Hence a retraction of a morphism is a left-inverse.
In this situation, is a split epimorphism and is a split monomorphism. The entire situation is said to be a splitting of the idempotent
Accordingly, a split monomorphism is a morphism that has a retraction; a split epimorphism is a morphism that is a retraction.
If an object has the left lifting property against a morphism , then so does every of its retracts :
This appears as (Borceux, lemma 6.5.6)
To the point
- In a category with terminal object every morphism of the form is a retract, and the unique morphism is the corresponding retraction.
The inclusion of standard topological horns into the topological simplex is a retract in Top.
In arrow categories
Let be the interval category. For every category the functor category is the arrow category of .
In the theory of weak factorization systems and model categories, an important role is played by retracts in , the arrow category of . Explicitly spelled out in terms of the original category , a morphism is a retract of a morphism if we have commutative squares
such that the top and bottom rows compose to identities.
Classes of morphisms in a category that are given by a left or right lifting property are preserved under retracts in the arrow category . In particular the defining classes of a model category are closed under retracts.
This is fairly immediate, a proof is made explicit here.
In every category the class of isomorphisms is preserved under retracts in the arrow category
This is also checked directly: for
a retract diagram and an isomorphism, the inverse to is given by the composite
where is the inverse of the middle morphism.
Retracts of diagrams
For the following, let and be categories and write for the join of with a single initial object, so that functors are precisely cones over functors . Write
for the canonical inclusion and hence for the underlying diagram of a cone . Finally, write for the functor category.
If is a retract in the category and is a limit cone over the diagram , then also is a limit cone over .
We give a direct and a more abstract argument.
Direct argument. We can directly check the universal property of the limit: for any other cone over , the composite exhibits also as a cone over . By the pullback property of this extends to a morphism of cones . Postcomposition with makes this a morphism of cones . By the injectivity of and the universality of , any two such cone morphisms are equals.
More abstract argument. The limiting cone over a diagram may be regarded as the right Kan extension along
Therefore a cone is limiting precisely if the -unit
is an isomorphism. Since this unit is a natural transformation it follows that applied to the retract diagram
it yields the retract diagram
in . Here by assumption the middle morphism is an isomorphism. Since isomorphisms are stable under retract, by prop. 4, also the left and right vertical morphism is an isomorphism, hence also is a limiting cone.
This argument generalizes form limits to homotopy limits.
For that, let now be a category with weak equivalences and write for the corresponding derivator: is the homotopy category of -diamgrams in , with respect to the degreewise weak equivalences in .
be a retract in . If is a homotopy limit cone over , then also is a homotopy limit cone over .
By the discussion at derivator we have that
forms homotopy limit cones;
is an isomorphism precisely if is a homotopy limit cone.
With this the claim follows as in prop. 5.
the definition appears as def. 1.7.3. Properties are discussed in section 6.5