nLab retract (Rev #23, changes)

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Context

Category theory

Idempotents

Definition
Properties
Examples
Related concepts
References

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 $\Delta[1] = \{0 \to 1\}$ 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

\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 $[\Delta[1],C]$ . 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:

Proposition

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

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 $[\Delta[1], [J^{\triangleleft}, C]]$ . 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 $Ho(C) : Diagram^{op} \to Cat$ 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

$i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})$ forms homotopy limit cones;
$F \to i_* i^* F$ is an isomorphism precisely if $F$ is a homotopy limit cone.

With this the claim follows as in prop. 5.

Related concepts

~~deformation retract~~ ,
deformation retract, neighbourhood retract
~~neighbourhood retract~~

absolute retract

References

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

Francis Borceux, Handbook of categorical algebra I

nLab retract (Rev #23, changes)

Context

Category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Idempotents

Examples

Contents

Definition

Properties

Lemma

Proof

Remark

Remark

Proposition

Proposition

Examples

To the point

Of simplices

In arrow categories

Proposition

Proposition

Proof

Retracts of diagrams

Proposition

Proof

Corollary

Proof

Related concepts

References