Delta Functors and Universality

By [L2], each degree of the horseshoe row is a split short exact sequence $0\to P_n^{\mathrm{\prime }}\to H_n\to P_n^{\mathrm{\prime }\mathrm{\prime }}\to 0.$ Because $F$ is additive by [L1], it preserves the biproduct decomposition carried by that split sequence, so each degree remains exact after applying $F$.

By [L3], the degreewise images from step 1.1 assemble into a sequence of chain maps $0\to F(P_{\bullet ,\mathrm{d}\mathrm{e}\mathrm{l}}^{\mathrm{\prime }})\to F(H_{\bullet ,\mathrm{d}\mathrm{e}\mathrm{l}})\to F(P_{\bullet ,\mathrm{d}\mathrm{e}\mathrm{l}}^{\mathrm{\prime }\mathrm{\prime }})\to 0.$ Since it is exact in every degree, [L4] identifies it as a short exact sequence of complexes.

By [L1], each horseshoe choice gives a short exact sequence of complexes after applying $F$, hence a homology connecting morphism. For a morphism of the underlying short exact sequences, fix comparison lifts on the two end resolutions. By [L3], write each horseshoe term as the biproduct of its end terms. Inductively in the degree, define the middle comparison map as the fixed diagonal pair of side maps plus an off-diagonal correction. The chain-map defect in degree $n$ lands in the kernel of the target augmentation or differential; projectivity of the corresponding right-hand resolution term lifts it, giving the next correction. This produces a middle chain map for which both side squares commute, hence a morphism of short exact sequences of complexes.

Applied to the identity morphism of the original short exact sequence, step 1.1 supplies a comparison between any two chosen horseshoes. By [L2], any two compatible middle comparisons with the fixed side maps are chain-homotopic.

Applying [L4] to the ladder from step 1.1 shows that the homology connecting morphisms commute with the induced maps on the two ends. If the end comparison lifts are changed, [L6] makes the old and new lifts chain-homotopic and [L5] makes their induced end homology maps equal. With either fixed pair of side lifts, [L2] likewise makes any two compatible middle maps chain-homotopic. Thus all transport maps occurring in the connecting square are unchanged on homology. Applying this to the comparison in step 2.1 shows that the resulting map $L_n^{P}F(A^{\mathrm{\prime }\mathrm{\prime }})\to L_{n-1}^{P}F(A^{\mathrm{\prime }})$ is independent of both the horseshoe and transport choices.

By [L2], choose any horseshoe middle resolution for the given short exact sequence and define the connecting maps from its long exact homology sequence. Using [L3], that horseshoe sequence yields an exact long sequence $\cdots \to L_n^{P}F(A^{\mathrm{\prime }})\to L_n^{P}F(A)\to L_n^{P}F(A^{\mathrm{\prime }\mathrm{\prime }})\stackrel{{\mathrm{\partial }}_n}{\to }{L}_{n-1}^{P}F(A^{\mathrm{\prime }})\to \cdots ,$ and item 10 ensures that this sequence depends only on the original short exact sequence, not on the auxiliary horseshoe data.

For a morphism of short exact sequences in $\mathcal{A}$, the generalized comparison assertion in [L2] supplies a compatible morphism between chosen horseshoe sequences. Apply [L3] to it. The connecting squares commute on the horseshoe level, and [L2] transports that naturality to the fixed datum $P$. Together with the additivity from [L1], this is exactly the homological delta-functor structure required by [L5].

The degree-zero term is naturally isomorphic to $F$ by [L4]. Hence the left derived functors form a homological delta functor with degree zero equal to the original right exact functor.

Choose the injective horseshoe from [L2]. Its degreewise split short exact sequence remains split exact after applying the additive functor $F$, so it becomes a short exact sequence of cochain complexes in $\mathcal{B}$. Now [L3] gives its long exact cohomology sequence, and [L4] transports the middle cohomology groups to the fixed datum $I$. This defines connecting maps ${\mathrm{\partial }}^n:R_I^{n}F(A^{\mathrm{\prime }\mathrm{\prime }})\to R_I^{n+1}F(A^{\mathrm{\prime }})\mathrm{.}$

Given a morphism of short exact sequences, use the degreewise biproduct form of the two injective horseshoes and construct compatible comparison maps by the dual comparison induction: injectivity extends the off-diagonal correction at each degree. By [L7], different comparison extensions induce the same maps on cohomology. Naturality of the cohomology connecting morphism in [L3] then makes the connecting squares commute, and [L1] supplies additivity. Therefore [L6] identifies $(R_I^{n}F,{\mathrm{\partial }}^n)$ as a cohomological delta functor on $\mathcal{A}$.

The degree-zero term is naturally isomorphic to $F$ by [L5].

By [L1], the components $L_n^P(\alpha )$ and $R_I^n(\alpha )$ are already natural in the object variable.

Compute the connecting maps for the chosen short exact sequence from a horseshoe resolution on the projective or injective side as in [L2]. Applying $\alpha$ degreewise gives a morphism between the two short exact sequences of complexes obtained after applying $F$ and $G$. By [L3], the corresponding connecting squares in homology or cohomology commute.

Step 2.1 is exactly the compatibility demanded in [L4]. Therefore the induced degreewise natural transformations from step 1.1 assemble into morphisms of the corresponding derived delta functors.

The first displayed sequence is exactly the long exact sequence attached by [L1] to the given short exact sequence.

The second displayed sequence is exactly the long exact sequence attached by [L2] to the same short exact sequence.

By [L1], choose a projective epimorphism $q:Q\twoheadrightarrow A$. The datum $P$ is defined on all of $\mathcal{A}$, so $L_n^{P}F(Q)$ is defined; since $Q$ is projective and $n>0$, [L2] gives $L_n^{P}F(Q)=0$.

The induced map $L_n^{P}F(q):L_n^{P}F(Q)\to L_n^{P}F(A)$ is therefore the zero map. By [L3], this is exactly the required positive-degree effacement of $L_n^{P}F(A)$ by a projective object.

By [L1], choose a monomorphism $u:A\rightarrowtail J$ with $J$ injective. The datum $I$ is defined on all of $\mathcal{A}$, so $R_I^{n}F(J)$ is defined; since $J$ is injective and $n>0$, [L2] gives $R_I^{n}F(J)=0$.

The induced map $R_I^{n}F(u):R_I^{n}F(A)\to R_I^{n}F(J)$ is therefore zero. By [L3], this is exactly the required positive-degree effacement.

By [L1], the kernel of ${\mathrm{\partial }}_n$ is the image of $T_n(P)\to T_n(A)$. Therefore if that incoming map is zero, then $\ker ({\mathrm{\partial }}_n)=0$ and ${\mathrm{\partial }}_n$ is monic.

Again by [L1], the image of ${\mathrm{\partial }}_n$ is the kernel of $T_{n-1}(K)\to T_{n-1}(P)$. If the outgoing map is zero, then that kernel is all of $T_{n-1}(K)$, so ${\mathrm{\partial }}_n$ is epic.

When both hypotheses hold, steps 1.1 and 1.2 show that ${\mathrm{\partial }}_n$ is both monic and epic, hence an isomorphism in the abelian target category.

By [L1], the kernel of ${\mathrm{\partial }}^{n-1}$ is the image of $T^{n-1}(I)\to T^{n-1}(C)$. If that map is zero, then $\ker ({\mathrm{\partial }}^{n-1})=0$, so ${\mathrm{\partial }}^{n-1}$ is monic.

By [L1], the image of ${\mathrm{\partial }}^{n-1}$ is the kernel of $T^n(A)\to T^n(I)$. If the latter map is zero, this kernel is all of $T^n(A)$, so ${\mathrm{\partial }}^{n-1}$ is epic.

When both hypotheses hold, steps 1.1 and 1.2 show that ${\mathrm{\partial }}^{n-1}$ is an isomorphism.

In the homological case, exactness of the long sequences for the chosen short exact sequence gives $S_n(P)\to S_n(A)\stackrel{{\mathrm{\partial }}_n^S}{\to }{S}_{n-1}(K)\to S_{n-1}(P)$ and $T_n(P)\to T_n(A)\stackrel{{\mathrm{\partial }}_n^T}{\to }{T}_{n-1}(K)\to T_{n-1}(P)\mathrm{.}$ Because the lower incoming map is zero by the chosen effacement, [L2] makes ${\mathrm{\partial }}_n^T$ monic. The already defined map $u_{n-1}(K)$ therefore determines at most one map $u_n^{(p)}(A)$ satisfying ${\mathrm{\partial }}_n^T{u}_n^{(p)}(A)=u_{n-1}(K){\mathrm{\partial }}_n^S,$ and exactness shows that the right-hand side lands in $\ker (T_{n-1}(K)\to T_{n-1}(P))=\mathrm{im}({\mathrm{\partial }}_n^T)$, so this map exists.

If $f:A\to A^{\mathrm{\prime }}$ is covered by a morphism of chosen effacement sequences, then the already defined degree-$(n-1)$ maps are natural by [L3]. Applying the defining equation from step 1.1 on both objects and using naturality of the connecting morphisms inside the two long exact sequences shows that ${\mathrm{\partial }}_n^T$ has the same composite with both $T_n(f)u_n^{(p)}(A)$ and $u_n^{(p^{\mathrm{\prime }})}(A^{\mathrm{\prime }})S_n(f)$. Since the target ${\mathrm{\partial }}_n^T$ is monic by [L2], these two maps are equal.

In the cohomological case, [L2] makes $q_S$ an isomorphism. The known map $u^n$ induces ${\bar{u}}^n$ on the displayed cokernels, while exactness makes ${\mathrm{\partial }}_T^n$ factor through the map $q_T$ from the target cokernel. Define $u_A^{n+1,(e)}:=q_T{\bar{u}}^n{q}_S^{-1}\mathrm{.}$ This is the unique map satisfying $u_A^{n+1,(e)}{q}_S=q_T{\bar{u}}^n$. The same cokernel equation, together with naturality of the known degree-$n$ maps from [L3], gives naturality for morphisms covered by chosen effacement ladders.

In the homological case, let $p_i:P_i\twoheadrightarrow A$ be two effacements of the relevant target value. Form the epimorphism $p=(p_1,p_2):P_1\oplus P_2\twoheadrightarrow A\mathrm{.}$ By [L2], $P_1\oplus P_2$ is projective. Since $T_n$ is additive, the canonical biproduct identification gives $T_n(P_1\oplus P_2)\cong T_n(P_1)\oplus T_n(P_2),$ and under this identification the map $T_n(p)$ has components $T_n(p_1)$ and $T_n(p_2)$, both zero. Thus $T_n(p)=0$, so $p$ is again an admissible effacement.

The inclusions ${\iota }_i:P_i\to P_1\oplus P_2$ satisfy $p{\iota }_i=p_i$, so they give morphisms from each original effacement to the dominating effacement of step 1.1 over the identity of $A$. By the naturality part of [L1], the component defined from $p_i$ agrees with the one defined from $p$ for each $i$. Hence the components defined from $p_1$ and $p_2$ are equal.

The cohomological case is dual: if $e_i:A\rightarrowtail I_i$ are two effacements, then $e=(e_1,e_2):A\rightarrowtail I_1\times I_2$ is again an admissible effacement by [L2], and the projections $I_1\times I_2\to I_i$ compare it with each original choice. Applying [L1] as in step 2.1 shows that the resulting degree-$(n+1)$ component is independent of the chosen injective effacement.

In the homological case, write the given sequence as $0\to A^{\mathrm{\prime }}\to A\to A^{\mathrm{\prime }\mathrm{\prime }}\to 0$ and choose the projective effacement $0\to K\to P\stackrel{p}{\to }{A}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ used by [L1] to define $u_n(A^{\mathrm{\prime }\mathrm{\prime }})$. Projectivity of $P$ lifts $p$ through the epimorphism $A\to A^{\mathrm{\prime }\mathrm{\prime }}$; the lift restricts to a map $k:K\to A^{\mathrm{\prime }}$, producing a morphism from the effacement sequence to the given sequence. By [L2], using this ladder-compatible effacement does not change $u_n(A^{\mathrm{\prime }\mathrm{\prime }})$. Naturality of the two connecting morphisms, the defining equation from [L1], and naturality of the already constructed $u_{n-1}$ give ${\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^T{u}_n(A^{\mathrm{\prime }\mathrm{\prime }})=T_{n-1}(k){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^T{u}_n(A^{\mathrm{\prime }\mathrm{\prime }})=T_{n-1}(k)u_{n-1}(K){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^S=u_{n-1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^S$. This is the required homological connecting square.

In the cohomological case, choose the injective effacement $0\to A^{\mathrm{\prime }}\stackrel{e}{\to }I\to C\to 0$ used by [L1] to define $u^{n+1}(A^{\mathrm{\prime }})$. Injectivity of $I$ extends $e$ across the monomorphism $A^{\mathrm{\prime }}\to A$ and induces a map $A^{\mathrm{\prime }\mathrm{\prime }}\to C$, producing a morphism from the given sequence to the effacement sequence. Again [L2] permits this compatible choice. Naturality of the connectors, the defining cokernel equation from [L1], and naturality of $u^n$ then give $u^{n+1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^S=u^{n+1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^S{S}^n(A^{\mathrm{\prime }\mathrm{\prime }}\to C)={\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^T{T}^n(A^{\mathrm{\prime }\mathrm{\prime }}\to C)u^n(A^{\mathrm{\prime }\mathrm{\prime }})={\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^T{u}^n(A^{\mathrm{\prime }\mathrm{\prime }})$. This is the required cohomological connecting square.

The two cases show that the one-step extension preserves every connecting morphism, independently of the effacement choices by [L2].

Start with the given $u_0$ as the degree-zero component.

Suppose by induction that for some $n>0$ we have already constructed natural transformations $u_i:S_i\Rightarrow T_i$ for all $i<n$, and that these form a morphism of homological delta functors through degree $n-1$. For each object $A$, choose an effacement $p_A:P_A\twoheadrightarrow A$ for the target value $T_n(A)$ using [L2]. Then [L3] defines a map $u_n(A):S_n(A)\to T_n(A)$ from that effacement, and [L3] makes it independent of the chosen $p_A$.

To check naturality of the family from step 1.2, compare two chosen effacements over a morphism $f:A\to B$ by a common dominating effacement. The covered naturality from [L3] applies to that dominating choice, and the choice independence from [L3] transports the result back to the original objects. Thus $u_n$ is natural. The same comparison argument, now applied over a short exact sequence, together with the connecting-map compatibility from [L3], shows that adjoining $u_n$ preserves the morphism-of-delta-functors condition in degree $n$.

This constructs a morphism $u:S\to T$ extending $u_0$ in every degree. For uniqueness, let $v_n$ be any other degree-$n$ component compatible with the already fixed lower-degree data. Choose an effacement $p:P\twoheadrightarrow A$ for $T_n(A)$. Because $u_n$ and $v_n$ have the same lower-degree compatibility, their composites with ${\mathrm{\partial }}_n^T$ agree. The map ${\mathrm{\partial }}_n^T$ is monic by [L2], so $u_n(A)=v_n(A)$. Hence the extension is unique in each degree, and [L1] identifies $T$ as universal.

Set the degree-zero component to be the given map $u^0$.

Suppose by induction that for some $n\ge 0$ we have already constructed natural transformations $u^i:T^i\Rightarrow S^i$ for all $i\le n$, compatible with the connecting morphisms through degree $n-1$. For each object $A$, choose an injective effacement $e_A:A\rightarrowtail I_A$ for $T^{n+1}(A)$ using [L2]. The cohomological clause of [L3], which is valid for every $n\ge 0$, defines a map $u_A^{n+1}:T^{n+1}(A)\to S^{n+1}(A)$, and [L3] makes it independent of the chosen effacement.

Naturality and compatibility with the connecting morphisms follow exactly as in the homological case: use the covered naturality from [L3] on a common dominating injective effacement and then appeal to the choice-independence from [L3]. Thus adjoining $u^{n+1}$ extends the partial morphism one degree further.

Uniqueness in degree $n+1$ comes from the cokernel description in [L2]: once the degree-$n$ component is fixed, item 19 gives only one possible map out of $T^{n+1}(A)$. Hence the inductive extension is unique in every degree, and [L1] shows that $T$ is universal.

In the right exact case, [L1] makes $(L_n^{P}F)$ a homological delta functor and [L2] makes it effaceable in positive degrees. Therefore [L3] shows that it is universal.

In the left exact case, [L1] makes $(R_I^{n}F)$ a cohomological delta functor and [L2] makes it positively effaceable. Applying [L3] again gives its universality.

In the homological case, set $t_0={\tau }^{-1}\sigma :S_0\to T_0$. Applying [L1] to $t_0$ gives a morphism $\varphi :S\to T$, and applying [L1] to $t_0^{-1}$ gives a morphism $\psi :T\to S$. Their composites extend ${\mathrm{id}}_{S_0}$ and ${\mathrm{id}}_{T_0}$ respectively, so uniqueness in [L1] forces $\psi \varphi ={\mathrm{id}}_S$ and $\varphi \psi ={\mathrm{id}}_T$. Hence $\varphi$ is the unique isomorphism extending ${\tau }^{-1}\sigma$.

The cohomological case is identical, with the maps oriented out of the universal functors as required by [L1].