Definition 12.12.1. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. A cohomological $\delta $-functor or simply a $\delta $-functor from $\mathcal{A}$ to $\mathcal{B}$ is given by the following data:
a collection $F^ n : \mathcal{A} \to \mathcal{B}$, $n \geq 0$ of additive functors, and
for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ a collection $\delta _{A \to B \to C} : F^ n(C) \to F^{n + 1}(A)$, $n \geq 0$ of morphisms of $\mathcal{B}$.
These data are assumed to satisfy the following axioms
for every short exact sequence as above the sequence
is exact, and
for every morphism $(A \to B \to C) \to (A' \to B' \to C')$ of short exact sequences of $\mathcal{A}$ the diagrams
are commutative.
Note that this in particular implies that $F^0$ is left exact.
Definition 12.12.2. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $(F^ n, \delta _ F)$ and $(G^ n, \delta _ G)$ be $\delta $-functors from $\mathcal{A}$ to $\mathcal{B}$. A morphism of $\delta $-functors from $F$ to $G$ is a collection of transformation of functors $t^ n : F^ n \to G^ n$, $n \geq 0$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ the diagrams are commutative.
\[ \xymatrix{ F^ n(C) \ar[d]_{t^ n} \ar[rr]_{\delta _{F, A \to B \to C}} & & F^{n + 1}(A) \ar[d]^{t^{n + 1}} \\ G^ n(C) \ar[rr]^{\delta _{G, A \to B \to C}} & & G^{n + 1}(A) } \]
Definition 12.12.3. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^ n, \delta _ F)$ be a $\delta $-functor from $\mathcal{A}$ to $\mathcal{B}$. We say $F$ is a universal $\delta $-functor if and only if for every $\delta $-functor $G = (G^ n, \delta _ G)$ and any morphism of functors $t : F^0 \to G^0$ there exists a unique morphism of $\delta $-functors $\{ t^ n\} _{n \geq 0} : F \to G$ such that $t = t^0$.
Lemma 12.12.4. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^ n, \delta _ F)$ be a $\delta $-functor from $\mathcal{A}$ to $\mathcal{B}$. Suppose that for every $n > 0$ and any $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ there exists an injective morphism $u : A \to B$ (depending on $A$ and $n$) such that $F^ n(u) : F^ n(A) \to F^ n(B)$ is zero. Then $F$ is a universal $\delta $-functor.
Proof. Let $G = (G^ n, \delta _ G)$ be a $\delta $-functor from $\mathcal{A}$ to $\mathcal{B}$ and let $t : F^0 \to G^0$ be a morphism of functors. We have to show there exists a unique morphism of $\delta $-functors $\{ t^ n\} _{n \geq 0} : F \to G$ such that $t = t^0$. We construct $t^ n$ by induction on $n$. For $n = 0$ we set $t^0 = t$. Suppose we have already constructed a unique sequence of transformation of functors $t^ i$ for $i \leq n$ compatible with the maps $\delta $ in degrees $\leq n$.
Let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. By assumption we may choose an embedding $u : A \to B$ such that $F^{n + 1}(u) = 0$. Let $C = B/u(A)$. The long exact cohomology sequence for the short exact sequence $0 \to A \to B \to C \to 0$ and the $\delta $-functor $F$ gives that $F^{n + 1}(A) = \mathop{\mathrm{Coker}}(F^ n(B) \to F^ n(C))$ by our choice of $u$. Since we have already defined $t^ n$ we can set
equal to the unique map such that
commutes. This is clearly uniquely determined by the requirements imposed. We omit the verification that this defines a transformation of functors. $\square$
Lemma 12.12.5. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. If there exists a universal $\delta $-functor $(F^ n, \delta _ F)$ from $\mathcal{A}$ to $\mathcal{B}$ with $F^0 = F$, then it is determined up to unique isomorphism of $\delta $-functors.
Proof. Immediate from the definitions. $\square$
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