An object $A$ in a category is called a retract of an object $B$ if there are morphisms $i\colon A\to B$ and $r \colon B\to A$ such that $r \circ i = id_A$. In this case $r$ is called a retraction of $B$ onto $A$.
Here $i$ may also be called a section of $r$. (In particular if $r$ is thought of as exhibiting a bundle; the terminology originates from topology.)
Hence a retraction of a morphism $i \;\colon\; A \to B$ is a left-inverse.
In this situation, $r$ is a split epimorphism and $i$ 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.
(left inverse with left inverse is inverse)
Let $\mathcal{C}$ be a category, and let $f$ and $g$ be morphisms in $\mathcal{C}$, such that $g$ is a left inverse to $f$:
If $g$ itself has a left inverse $h$
then $h = f$ and $g = f^{-1}$ is an actual (two-sided) inverse morphism to $f$.
Since inverse morphisms are unique if they exists, it is sufficient to show that
Compute as follows:
Retracts are clearly preserved by any functor.
A split epimorphism $r; B \to A$ is the strongest of various notions of epimorphism (e.g., it is a regular epimorphism, in fact an absolute? coequalizer, being the coequalizer of a pair $(e, 1_B)$ where $e = i \circ r: B \to B$ is idempotent). Dually, a split monomorphism is the strongest of various notions of monomorphism.
If an object $B$ has the left lifting property against a morphism $X \to Y$, then so does every of its retracts $A \to B$:
Let $C$ be a category with split idempotents and write $PSh(C) = [C^{op}, Set]$ for its presheaf category. Then a retract of a representable functor $F = PSh(C)$ is itself representable.
This appears as (Borceux, lemma 6.5.6)
The inclusion of standard topological horns into the topological simplex $\Lambda^n_k \hookrightarrow \Delta^n$ is a retract in Top.
Let $\Delta[1] = \{0 \to 1\}$ be the interval category. For every category $C$ the functor category $[\Delta[1], C]$ is the arrow category of $C$.
In the theory of weak factorization systems and model categories, an important role is played by retracts in $C^{\Delta[1]}$, the arrow category of $C$. Explicitly spelled out in terms of the original category $C$, a morphism $f:X\to Y$ is a retract of a morphism $g:Z\to W$ if we have commutative squares
such that the top and bottom rows compose to identities.
Classes of morphisms in a category $C$ that are given by a left or right lifting property are preserved under retracts in the arrow category $[\Delta[1],C]$. In particular the defining classes of a model category are closed under retracts.
This is fairly immediate, a proof is made explicit here.
This implies:
In every category $C$ the class of isomorphisms is preserved under retracts in the arrow category $[\Delta[1], C]$
This is also checked directly: for
a retract diagram and $a_2 \to b_2$ an isomorphism, the inverse to $a_1 \to b_1$ is given by the composite
where $b_2 \to a_2$ is the inverse of the middle morphism.
For the following, let $C$ and $J$ be categories and write $J^{\triangleleft}$ for the join of $J$ with a single initial object, so that functors $J^{\triangleleft} \to C$ are precisely cones over functors $J \to C$. Write
for the canonical inclusion and hence $i^* F$ for the underlying diagram of a cone $F : J^{\triangleleft} \to C$. Finally, write $[J^{\triangleleft}, C]$ for the functor category.
If $Id: F_1 \hookrightarrow F_2 \to F_1$ is a retract in the category $[J^{\triangleleft}, C]$ and $F_2 : J^{\triangleleft} \to C$ is a limit cone over the diagram $i^* F_2 : J \to C$, then also $F_1$ is a limit cone over $i^* F_1$.
We give a direct and a more abstract argument.
Direct argument. We can directly check the universal property of the limit: for $G$ any other cone over $i^* F_1$, the composite $i^* G = i^* F_1 \to i^* F_2$ exhibits $G$ also as a cone over $i^* F_2$. By the pullback property of $F_2$ this extends to a morphism of cones $G \to F_2$. Postcomposition with $F_2 \to F_1$ makes this a morphism of cones $G \to F_1$. By the injectivity of $F_1 \to F_2$ and the universality of $F_2$, any two such cone morphisms are equals.
More abstract argument. The limiting cone over a diagram $D : J \to C$ may be regarded as the right Kan extension $i_* D := Ran_i D$ along $i$
Therefore a cone $F : J^{\triangleleft} \to C$ is limiting precisely if the $(i^* \dashv i_*)$-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 $[\Delta[1], [J^{\triangleleft}, C]]$. Here by assumption the middle morphism is an isomorphism. Since isomorphisms are stable under retract, by prop. , also the left and right vertical morphism is an isomorphism, hence also $F_1$ is a limiting cone.
This argument generalizes form limits to homotopy limits.
For that, let now $C$ be a category with weak equivalences and write $Ho(C) : Diagram^{op} \to Cat$ for the corresponding derivator: $Ho(C)(J) := [J,C](W^J)^{-1}$ is the homotopy category of $J$-diagrams in $C$, with respect to the degreewise weak equivalences in $C$.
Let
be a retract in $Ho(C)(J^{\triangleleft})$. If $F_2$ is a homotopy limit cone over $i^* F_2$, then also $F_1$ is a homotopy limit cone over $i^* F_1$.
By the discussion at derivator we have that
$i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})$ forms homotopy limit cones;
$F \to i_* i^* F$ is an isomorphism precisely if $F$ is a homotopy limit cone.
Karol Borsuk, Theory of retracts
Sibe Mardešić, Absolute Neighborhood Retracts and Shape Theory (pdf)
Francis Borceux, Def. 1.7.3 and Sec. 6.5 of: Handbook of Categorical Algebra I
See also