nLab
injective or projective morphism

Contents

Contents

Idea

Injective and projective morphisms are morphisms in a category satisfying some lifting property. Often they appear jointly to form weak factorization systems.

Definition

Injective and projective

Given a category 𝒞\mathcal{C} and a class JMor(𝒞)J \subset Mor(\mathcal{C}) of its morphisms, then any morphism fMor(𝒞)f \in Mor(\mathcal{C}) is called

If 𝒞\mathcal{C} has a terminal object *\ast, then an object XX for which X!*X \stackrel{\exists!}{\to} \ast is a JJ-injective morphism is called a JJ-injective object.

If 𝒞\mathcal{C} has an initial object \emptyset, then an object XX for which !X\emptyset \stackrel{\exists!}{\to} X is a JJ-projective morphism is called a JJ-projective object.

Fibration and cofibration

Frequently one furthermore says:

Properties

Closure properties

Proposition

Let 𝒞\mathcal{C} be a category and let KMor(𝒞)K\subset Mor(\mathcal{C}) be a class of morphisms. Write KProjK Proj and KInjK Inj, respectively, for the sub-classes of KK-projective morphisms and of KK-injective morphisms. Then:

  1. Both classes contain the class of isomorphism of 𝒞\mathcal{C}.

  2. Both classes are closed under composition in 𝒞\mathcal{C}.

    KProjK Proj is also closed under transfinite composition.

  3. Both classes are closed under forming retracts in the arrow category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}.

  4. KProjK Proj is closed under forming pushouts of morphisms in 𝒞\mathcal{C} (“cobase change”).

    KInjK Inj is closed under forming pullback of morphisms in 𝒞\mathcal{C} (“base change”).

  5. KProjK Proj is closed under forming coproducts in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}.

    KInjK Inj is closed under forming products in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}.

Proof

We go through each item in turn.

containing isomorphisms

Given a commuting square

A f X Iso i p B g Y \array{ A &\overset{f}{\rightarrow}& X \\ {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}} \\ B &\underset{g}{\longrightarrow}& Y }

with the left morphism an isomorphism, the a lift is given by using the inverse of this isomorphism fi 1{}^{{f \circ i^{-1}}}\nearrow. Hence in particular there is a lift when pKp \in K and so iKProji \in K Proj. The other case is formally dual.

closure under composition

Given a commuting square of the form

A X KInj p 1 K i KInj p 2 B Y \array{ A &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }

consider its pasting decomposition as

A X KInj p 1 K i KInj p 2 B Y. \array{ A &\longrightarrow& X \\ \downarrow &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } \,.

Now the bottom commuting square has a lift, by assumption. This yields another pasting decomposition

A X K i KInj p 1 KInj p 2 B Y \array{ A &\longrightarrow& X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that p 1p 1p_1\circ p_1 has the right lifting property against KK and is hence in KInjK Inj. The case of composing two morphisms in KProjK Proj is formally dual. From this the closure of KProjK Proj under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property.

closure under retracts

Let jj be the retract of an iKProji \in K Proj, i.e. let there be a commuting diagram of the form.

id A: A C A j KProj i j id B: B D B. \array{ id_A \colon & A &\longrightarrow& C &\longrightarrow& A \\ & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} \\ id_B \colon & B &\longrightarrow& D &\longrightarrow& B } \,.

Then for

A X j K f B Y \array{ A &\longrightarrow& X \\ {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& Y }

a commuting square, it is equivalent to its pasting composite with that retract diagram

A C A X j KProj i j K f B D B Y. \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.

Now the pasting composite of the two squares on the right has a lift, by assumption,

A C A X j KProj i K f B D B Y. \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \nearrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence jj has the left lifting property against all pKp \in K and hence is in KProjK Proj. The other case is formally dual.

closure under pushout and pullback

Let pKInjp \in K Inj and and let

Z× fX X f *p p Z f Y \array{ Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}} \\ Z &\stackrel{f}{\longrightarrow} & Y }

be a pullback diagram in 𝒞\mathcal{C}. We need to show that f *pf^* p has the right lifting property with respect to all iKi \in K. So let

A Z× fX K i f *p B g Z \array{ A &\longrightarrow& Z \times_f X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}} \\ B &\stackrel{g}{\longrightarrow}& Z }

be a commuting square. We need to construct a diagonal lift of that square. To that end, first consider the pasting composite with the pullback square from above to obtain the commuting diagram

A Z× fX X i f *p p B g Z f Y. \array{ A &\longrightarrow& Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.

By the right lifting property of pp, there is a diagonal lift of the total outer diagram

A X i (fg)^ p B fg Y. \array{ A &\longrightarrow& X \\ \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{f g}{\longrightarrow}& Y } \,.

By the universal property of the pullback this gives rise to the lift g^\hat g in

Z× fX X g^ f *p p B g Z f Y. \array{ && Z \times_f X &\longrightarrow& X \\ &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.

In order for g^\hat g to qualify as the intended lift of the total diagram, it remains to show that

A Z× fX i g^ B \array{ A &\longrightarrow& Z \times_f X \\ \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow \\ B }

commutes. To do so we notice that we obtain two cones with tip AA:

  • one is given by the morphisms

    1. AZ× fXXA \to Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z

    with universal morphism into the pullback being

    • AZ× fXA \to Z \times_f X
  • the other by

    1. AiBg^Z× fXXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z.

    with universal morphism into the pullback being

    • AiBg^Z× fXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

The other case is formally dual.

closure under (co-)products

Let {(A si sB s)KProj} sS\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S} be a set of elements of KProjK Proj. Since colimits in the presheaf category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects sSA s\underset{s \in S}{\coprod} A_s induced via its universal property by the set of morphisms i si_s:

sSA s(i s) sSsSB s. \underset{s \in S}{\sqcup} A_s \overset{(i_s)_{s\in S}}{\longrightarrow} \underset{s \in S}{\sqcup} B_s \,.

Now let

sSA s X (i s) sS K f sSB s Y \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y }

be a commuting square. This is in particular a cocone under the coproduct of objects, hence by the universal property of the coproduct, this is equivalent to a set of commuting diagrams

{A s X KProj i s K f B s Y} sS. \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in K Proj}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} \,.

By assumption, each of these has a lift s\ell_s. The collection of these lifts

{A s X Proj i s s K f B s Y} sS \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in Proj}}\downarrow &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S}

is now itself a compatible cocone, and so once more by the universal property of the coproduct, this is equivalent to a lift ( s) sS(\ell_s)_{s\in S} in the original square

sSA s X (i s) sS ( s) sS K f sSB s Y. \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow &{}^{(\ell_s)_{s\in S}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } \,.

This shows that the coproduct of the i si_s has the left lifting property against all fKf\in K and is hence in KProjK Proj. The other case is formally dual.

  • the small object argument is about factoring morphisms into JJ-cofibrations followed by JJ-injective morphisms.

Last revised on April 22, 2016 at 09:30:55. See the history of this page for a list of all contributions to it.