nLab retract




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

id:AsectioniBretractionrA. 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:ABi \;\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

BrAiB. 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.



(left inverse with left inverse is inverse)

Let 𝒞\mathcal{C} be a category, and let ff and gg be morphisms in 𝒞\mathcal{C}, such that gg is a left inverse to ff:

gf=id. g \circ f = id \,.

If gg itself has a left inverse hh

hg=id h \circ g = id

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


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

fg=id. f \circ g = id \,.

Compute as follows:

fg =hg=idfg =hgf=idg =hg =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}

Retracts are clearly preserved by any functor.


A split epimorphism r;BAr; 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)(e, 1_B) where e=ir:BBe = i \circ r: B \to B is idempotent). Dually, a split monomorphism is the strongest of various notions of monomorphism.


If an object BB has the left lifting property against a morphism XYX \to Y, then so does every of its retracts ABA \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)

Let CC be a category with split idempotents and write PSh(C)=[C op,Set]PSh(C) = [C^{op}, Set] for its presheaf category. Then a retract of a representable functor F=PSh(C)F = PSh(C) is itself representable.

This appears as (Borceux, lemma 6.5.6)


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 nΔ n\Lambda^n_k \hookrightarrow \Delta^n is a retract in Top.

In arrow categories

Let Δ[1]={01}\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:XYf:X\to Y is a retract of a morphism g:ZWg: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.


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:


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


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 2b 2a_2 \to b_2 an isomorphism, the inverse to a 1b 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 2a 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 JCJ \to C. Write

i:JJ 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.


If Id:F 1F 2F 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:JCi^* F_2 : J \to C, then also F 1F_1 is a limit cone over i *F 1i^* F_1.


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 1i *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 GF 2G \to F_2. Postcomposition with F 2F 1F_2 \to F_1 makes this a morphism of cones GF 1G \to F_1. By the injectivity of F 1F 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:JCD : 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

Fi *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 1F 2F 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 opCatHo(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.



Id:F 1F 2F 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.


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. Fi *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. .


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.