nLab
retract

Definition

An object $A$ in a category is a retract of an object $B$ if there are morphisms $i:A\to B$ and $r:B\to A$ such that $r\circ i={\mathrm{id}}_A$. In this case $r$ is called a retraction of $B$ onto $A$.

$$\mathrm{id}:A\stackrel{i}{\to }B\underset{\mathrm{retraction}}{\overset{r}{\to }}A.$$id : A \stackrel{i}{\to} B \underoverset{retraction}{r}{\to} A \,.

In this situation, $r$ is a split epimorphism and $i$ is a split monomorphism; the composite $i\circ r$ is a split idempotent. Sometimes $r$ is called a retraction of $i$ and $i$ is called a section of $r$; these terms come from topology. The whole thing may also be called a splitting of $i$, $r$, or $i\circ r$.

Properties

General

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 $A\to B$:

$$\left(\begin{array}{ccc}& & Y\\ & {}^{\exists }\nearrow & \downarrow \\ A& \to & Y\end{array}\right):=\left(\begin{array}{ccccccc}& & & & & & Y\\ & & & {}^{\exists }\nearrow & & & \downarrow \\ A& \to & B& \to & A& \to & Y\end{array}\right)$$\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 $\mathrm{PSh}(C)=[C^{\mathrm{op}},\mathrm{Set}]$ for its presheaf category. Then a retract of a representable functor $F=\mathrm{PSh}(C)$ is itself representable.

This appears as (Borceux, lemma 6.5.6)

Retracts of morphisms

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 $C$.

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 $[\Delta [1],C]$. In particular the defining classes of a model category are closed under retracts.

This implies:

Proposition

In every category $C$ the class of isomorphisms is preserved under retracts in $[\Delta [1],C]$

Proof

This is also checked directly: for

$$\begin{array}{cccccc}\mathrm{Id}:& a_1& \to & a_2& \to & a_1\\ & \downarrow & & \downarrow & & \downarrow \\ \mathrm{Id}:& b_1& \to & b_2& \to & b_1\end{array}$$\array{ Id: & a_1 &\to& a_2 &\to& a_1 \\ & \downarrow && \downarrow && \downarrow \\ Id: & 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

$$\begin{array}{cccccc}& & & a_2& \to & a_1\\ & & & \uparrow & & \\ & b_1& \to & b_2& & \end{array},$$\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 $J^{◃}$ for the join of $J$ with a single initial object, so that functors $J^{◃}\to C$ are precisely cones over functors $J\to C$. Write

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

for the canonical inclusion and hence $i^{*}F$ for the underlying diagram of a cone $F:J^{◃}\to C$. Finally, write $[J^{◃},C]$ for the functor category.

Proposition

If $\mathrm{Id}:F_1\hookrightarrow F_2\to F_1$ is a retract in the category $[J^{◃},C]$ and $F_2:J^{◃}\to C$ 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 $G\to F_2$. Postcompisition 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 $i_{*}D:={\mathrm{Ran}}_{i}D$ along $i$

$$\begin{array}{ccc}J& \stackrel{D}{\to }& C\\ {}^i\downarrow & {\nearrow }_{i_{*}D}\\ J^{◃}\end{array}.$$\array{ J &\stackrel{D}{\to}& C \\ {}^{\mathllap{i}}\downarrow & \nearrow_{i_* D} \\ J^{\triangleleft} } \,.

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

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

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

$$\mathrm{Id}:F_1\hookrightarrow F_2\to F_1$$Id : F_1 \hookrightarrow F_2 \to F_1

it yields the retract diagram

$$\begin{array}{cccccc}\mathrm{Id}:& F_1& \to & F_2& \to & F_1\\ & \downarrow & & \downarrow & & \downarrow \\ \mathrm{Id}:& i_{*}{i}^{*}{F}_1& \to & i_{*}{i}^{*}{F}_2& \to & i_{*}{i}^{*}{F}_1\end{array}$$\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 $[\Delta [1],[J^{◃},C]]$. 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 $F_1$ 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 $\mathrm{Ho}(C):{\mathrm{Diagram}}^{\mathrm{op}}\to \mathrm{Cat}$ for the corresponding derivator: $\mathrm{Ho}(C)(J):=[J,C](W^J{)}^{-1}$ is the homotopy category of $J$-diamgrams in $C$, with respect to the degreewise weak equivalences in $C$.

Corollary

Let

$$\mathrm{Id}:F_1\to F_2\to F_1$$Id : F_1 \to F_2 \to F_1

be a retract in $\mathrm{Ho}(C)(J^{◃})$. 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. $i_{*}:\mathrm{Ho}(C)(J)\to \mathrm{Ho}(C)(J^{◃})$ 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. 5.

Examples

General

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

Of simplices

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

In arrow categories

In the theory of weak factorization systems and model categories, an important role is played by retracts in $C^2$, 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

$$\begin{array}{ccccc}X& \to & Z& \to & X\\ f\downarrow & & g\downarrow & & \downarrow f\\ Y& \to & W& \to & Y\end{array}$$\array{X & \to & Z & \to & X\\ f \downarrow & & g \downarrow & & \downarrow f\\ Y & \to & W & \to & Y}

such that the top and bottom rows compose to identities.

Related concepts

• deformation retract, neighbourhood retract

References

In

• Francis Borceux, Handbook of categorical algebra I

the definition appears as def. 1.7.3. Properties are discussed in section 6.5