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retract in nLab

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Definition

An object AA in a category is called a retract of an object BB if there are morphisms i:A→Bi\colon A\to B and r:B→Ar \colon B\to A such that r∘i=id Ar \circ i = id_A. In this case rr is called a retraction of BB onto AA.

id:A→sectioniB→retractionrA. id \;\colon\; A \underoverset{section}{i}{\to} B \underoverset{retraction}{r}{\to} A \,.

Here ii may also be called a section of rr. (In particular if rr is thought of as exhibiting a bundle; the terminology originates from topology.)

Hence a retraction of a morphism i:A→Bi \;\colon\; A \to B is a left-inverse.

In this situation, rr is a split epimorphism and ii is a split monomorphism. The entire situation is said to be a splitting of the idempotent

B⟶rA⟶iB. 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

Proof

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

f∘g=id. f \circ g = id \,.

Compute as follows:

f∘g =h∘g⏟=id∘f∘g =h∘g∘f⏟=id∘g =h∘g =id \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}

Proposition

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

( X ∃↗ ↓ A → Y):=( X ∃↗ ↓ A → B → A → Y) \left( \array{ && X \\ & {}^{\mathllap{\exists}}\nearrow& \downarrow \\ A &\to& Y } \right) \;\;\;\; := \;\;\;\; \left( \array{ && && && X \\ &&& {}^{\mathllap{\exists}}\nearrow& && \downarrow \\ A &\to& B &\to& A &\to& Y } \right)

This appears as (Borceux, lemma 6.5.6)

Examples

To the point

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

Of simplices

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

In arrow categories

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

In the theory of weak factorization systems and model categories, an important role is played by retracts in C Δ[1]C^{\Delta[1]}, the arrow category of CC. Explicitly spelled out in terms of the original category CC, a morphism f:X→Yf:X\to Y is a retract of a morphism g:Z→Wg:Z\to W if we have commutative squares

id X: X → Z → X f↓ g↓ ↓f id Y: Y → W → Y \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 CC that are given by a left or right lifting property are preserved under retracts in the arrow category [Δ[1],C][\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 CC the class of isomorphisms is preserved under retracts in the arrow category [Δ[1],C][\Delta[1], C]

Proof

This is also checked directly: for

id: a 1 → a 2 → a 1 ↓ ↓ ↓ id: b 1 → b 2 → b 1 \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→b 2a_2 \to b_2 an isomorphism, the inverse to a 1→b 1a_1 \to b_1 is given by the composite

a 2 → a 1 ↑ b 1 → b 2 , \array{ & & & a_2 &\to& a_1 \\ & && \uparrow && \\ & b_1 &\to& b_2 && } \,,

where b 2→a 2b_2 \to a_2 is the inverse of the middle morphism.

Retracts of diagrams

For the following, let CC and JJ be categories and write J ◃J^{\triangleleft} for the join of JJ with a single initial object, so that functors J ◃→CJ^{\triangleleft} \to C are precisely cones over functors J→CJ \to C. Write

i:J→J ◃ i : J \to J^{\triangleleft}

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

Proposition

If Id:F 1↪F 2→F 1Id: F_1 \hookrightarrow F_2 \to F_1 is a retract in the category [J ◃,C][J^{\triangleleft}, C] and F 2:J ◃→CF_2 : J^{\triangleleft} \to C is a limit cone over the diagram i *F 2:J→Ci^* F_2 : J \to C, then also F 1F_1 is a limit cone over i *F 1i^* 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 GG any other cone over i *F 1i^* F_1, the composite i *G=i *F 1→i *F 2i^* G = i^* F_1 \to i^* F_2 exhibits GG also as a cone over i *F 2i^* F_2. By the pullback property of F 2F_2 this extends to a morphism of cones G→F 2G \to F_2. Postcomposition with F 2→F 1F_2 \to F_1 makes this a morphism of cones G→F 1G \to F_1. By the injectivity of F 1→F 2F_1 \to F_2 and the universality of F 2F_2, any two such cone morphisms are equals.

More abstract argument. The limiting cone over a diagram D:J→CD : J \to C may be regarded as the right Kan extension i *D:=Ran iDi_* D := Ran_i D along ii

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

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

F→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

Id:F 1↪F 2→F 1 Id : F_1 \hookrightarrow F_2 \to F_1

it yields the retract diagram

Id: F 1 → F 2 → F 1 ↓ ↓ ↓ Id: i *i *F 1 → i *i *F 2 → i *i *F 1 \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 [Δ[1],[J ◃,C]][\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 1F_1 is a limiting cone.

This argument generalizes form limits to homotopy limits.

For that, let now CC be a category with weak equivalences and write Ho(C):Diagram op→CatHo(C) : Diagram^{op} \to Cat for the corresponding derivator: Ho(C)(J):=[J,C](W J) −1Ho(C)(J) := [J,C](W^J)^{-1} is the homotopy category of JJ-diagrams in CC, with respect to the degreewise weak equivalences in CC.

Corollary

Let

Id:F 1→F 2→F 1 Id : F_1 \to F_2 \to F_1

be a retract in Ho(C)(J ◃)Ho(C)(J^{\triangleleft}). If F 2F_2 is a homotopy limit cone over i *F 2i^* F_2, then also F 1F_1 is a homotopy limit cone over i *F 1i^* F_1.

Proof

By the discussion at derivator we have that

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

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

With this the claim follows as in prop. .

References

See also

Last revised on October 13, 2023 at 15:09:24. See the history of this page for a list of all contributions to it.