Contents

category theory

# Contents

## Definition

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 $A$.

$id \;\colon\; A \underoverset{section}{i}{\to} B \underoverset{retraction}{r}{\to} A \,.$

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

$B \stackrel{r}{\longrightarrow} A \stackrel{i}{\longrightarrow} B \,.$

Accordingly, a split monomorphism is a morphism that has a retraction; a split epimorphism is a morphism that is a retraction.

## Properties

###### Lemma

(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 $f$:

$g \circ f = id \,.$

If $g$ itself has a left inverse $h$

$h \circ g = id$

then $h = f$ and $g = f^{-1}$ is an actual (two-sided) inverse morphism to $f$.

###### Proof

Since inverse morphisms are unique if they exists, it is sufficient to show that

$f \circ g = id \,.$

Compute as follows:

\begin{aligned} f \circ g & = \underset{ = id}{\underbrace{h \circ g}} \circ f \circ g \\ & = h \circ \underset{= id}{\underbrace{g \circ f}} \circ g \\ & = h \circ g \\ & = id \end{aligned}
###### Remark

Retracts are clearly preserved by any functor.

###### Remark

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.

###### Proposition

If an object $B$ has the left lifting property against a morphism $X \to Y$, then so does every of its retracts $A \to B$:

$\left( \array{ && Y \\ & {}^{\mathllap{\exists}}\nearrow& \downarrow \\ A &\to& Y } \right) \;\;\;\; := \;\;\;\; \left( \array{ && && && Y \\ &&& {}^{\mathllap{\exists}}\nearrow& && \downarrow \\ A &\to& B &\to& A &\to& Y } \right)$
###### Proposition

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)

## Examples

### To the point

• In a category with terminal object $*$ every morphism of the form $* \to X$ is a section, and the unique morphism $X \to *$ is the corresponding retraction.

### Of simplices

The inclusion of standard topological horns into the topological simplex $\Lambda^n_k \hookrightarrow \Delta^n$ is a retract in Top.

### In arrow categories

Let $\Delta = \{0 \to 1\}$ be the interval category. For every category $C$ the functor category $[\Delta, C]$ is the arrow category of $C$.

In the theory of weak factorization systems and model categories, an important role is played by retracts in $C^{\Delta}$, 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

$\array{ id_X \colon & X & \to & Z & \to & X \\ & f \downarrow & & g \downarrow & & \downarrow f \\ id_Y \colon & Y & \to & W & \to & Y }$

such that the top and bottom rows compose to identities.

###### Proposition

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 $[\Delta,C]$. 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:

###### Proposition

In every category $C$ the class of isomorphisms is preserved under retracts in the arrow category $[\Delta, C]$

###### Proof

This is also checked directly: for

$\array{ id \colon & a_1 &\to& a_2 &\to& a_1 \\ & \downarrow && \downarrow && \downarrow \\ id \colon & b_1 &\to& b_2 &\to& b_1 }$

a retract diagram and $a_2 \to b_2$ an isomorphism, the inverse to $a_1 \to b_1$ is given by the composite

$\array{ & & & a_2 &\to& a_1 \\ & && \uparrow && \\ & b_1 &\to& b_2 && } \,,$

where $b_2 \to a_2$ is the inverse of the middle morphism.

### Retracts of diagrams

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

$i : J \to J^{\triangleleft}$

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.

###### Proposition

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 $i^* F_1$.

###### Proof

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 $i$

$\array{ J &\stackrel{D}{\to}& C \\ {}^{\mathllap{i}}\downarrow & \nearrow_{i_* D} \\ J^{\triangleleft} } \,.$

Therefore a cone $F : J^{\triangleleft} \to C$ is limiting precisely if the $(i^* \dashv i_*)$-unit

$F \stackrel{}{\to} i_* i^* F$

is an isomorphism. Since this unit is a natural transformation it follows that applied to the retract diagram

$Id : F_1 \hookrightarrow F_2 \to F_1$

it yields the retract diagram

$\array{ Id : & F_1 &\to& F_2 &\to& F_1 \\ & \downarrow && \downarrow && \downarrow \\ Id : & i_* i^* F_1 &\to& i_* i^* F_2 &\to& i_* i^* F_1 }$

in $[\Delta, [J^{\triangleleft}, C]]$. 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 $J$-diamgrams in $C$, with respect to the degreewise weak equivalences in $C$.

###### Corollary

Let

$Id : F_1 \to F_2 \to F_1$

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 $i^* F_1$.

###### Proof

By the discussion at derivator we have that

1. $i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})$ forms homotopy limit cones;

2. $F \to i_* i^* F$ is an isomorphism precisely if $F$ is a homotopy limit cone.

With this the claim follows as in prop. .