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:

( Y A Y):=( Y A B A Y) \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)

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. 4, 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-diamgrams 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. 5.



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

Last revised on June 28, 2018 at 07:27:10. See the history of this page for a list of all contributions to it.