Low-degree exact sequences for Yoneda Ext in both variables

Every displayed map has the asserted direction: Hom uses composition; the two maps between short extension classes are $i_{\ast }$ and $p_{\ast }$; $\delta (f)=f^{\ast }E$; and $\mathrm{\partial }[F]=[EF]$. Universal properties send endpoint-fixed maps to endpoint-fixed maps, so pullback, pushout, and splicing respect the generated equivalence relation. These constructions commute with finite direct sums. Pullbacks and pushouts in independent endpoints commute, as is seen by their common universal property. Consequently the maps between short extension classes preserve Baer addition.

Exactness at the first three Hom terms follows from the kernel property of $i$ and the cokernel property of $p$: a morphism $X\to B$ is killed by $p$ exactly when it factors uniquely through $i$. At $\mathrm{Hom}(X,C)$, a map $f$ belongs to $\ker \delta$ exactly when $f^{\ast }E$ splits. A section of that pullback is exactly a map $s:X\to B$ satisfying $ps=f$. Thus $\ker \delta =\mathrm{im}(p\circ -)$.

The connecting maps are additive as well. For $f,g:X\to C$, form the pullback of $E\oplus E$ along $(f,g):X\to C\oplus C$ and then push out along $A\oplus A\stackrel{\mathrm{\nabla }}{\to }A$. Addition $B\oplus B\to B$ induces an endpoint-fixed map from this short extension to $(f+g{)}^{\ast }E$, proving $\delta (f+g)=\delta (f)+\delta (g)$. For the other connecting map, use the two-fold Baer construction: direct sum, diagonal pullback at $X$, codiagonal pushout at $A$. Write $F_0={\mathrm{\Delta }}_X^{\ast }(F\oplus F^{\mathrm{\prime }})$, whose kernel is $C\oplus C$. Then $F+F^{\mathrm{\prime }}={\mathrm{\nabla }}_{C\ast }{F}_0$. Put $H=B{\times }_C(C\oplus C)$, with pullback along ${\mathrm{\nabla }}_C$. The map $B\oplus B\to H$ induced by ${\mathrm{\nabla }}_B$ and $p\oplus p$ induces an isomorphism ${\mathrm{\nabla }}_{A\ast }(E\oplus E)\cong {\mathrm{\nabla }}_C^{\ast }E$ of extensions of $C\oplus C$ by $A$. Finally the defining pushout map $F_0\to {\mathrm{\nabla }}_{C\ast }{F}_0$ and pullback map ${\mathrm{\nabla }}_C^{\ast }E\to E$ give an endpoint-fixed chain map between their splices. These explicit maps identify the two-fold Baer representative of $[EF]+[E{F}^{\mathrm{\prime }}]$ with $[E(F+F^{\mathrm{\prime }})]$. Thus $\mathrm{\partial }$ preserves addition wherever the extension-class collections are sets; the same formulas hold on classes.

Let $H:0\to A\stackrel{a}{\to }T\stackrel{t}{\to }X\to 0$. If $i_{\ast }H$ splits, its middle pushout $P=B{⨿}_{A}T$ admits a retraction to its subobject $B$. Composing $T\to P$ with this retraction gives $h:T\to B$ with $ha=i$. Hence $ph$ factors uniquely through $t$ as $f:X\to C$. The pair $(h,t)$ gives an endpoint-fixed map $H\to f^{\ast }E$, so $[H]=\delta (f)$. Conversely $i_{\ast }E$ splits: on its pushout $B{⨿}_{A}B$, the two identity maps to $B$ induce a retraction onto the first copy of $B$. Pulling back along $f$ shows $i_{\ast }\delta (f)=0$. This proves exactness at ${\mathrm{YExt}}^1(X,A)$.

Let $H:0\to B\stackrel{b}{\to }T\stackrel{t}{\to }X\to 0$ and suppose $p_{\ast }H$ splits. A retraction from its middle pushout to $C$, composed with $T$'s pushout map, yields $h:T\to C$ satisfying $hb=p$. Let $K=\ker h$. The kernel of $t{\mathrm{\mid }}_K$ is $A$, and $t{\mathrm{\mid }}_K:K\to X$ is epic. For the last assertion without elements, if $z:X\to Y$ satisfies $zt{\mathrm{\mid }}_K=0$, then $zt$ factors through $h$, since $h$ is epic and $K=\ker h$. Write $zt=wh$; restricting to $B$ gives $wp=0$, so $w=0$ and then $z=0$. Thus the cokernel of $t{\mathrm{\mid }}_K$ is zero. The resulting extension $0\to A\to K\to X\to 0$ pushes out along $i$ to $H$: the map $B{⨿}_{A}K\to T$ induced by $b$ and the inclusion of $K$ is an endpoint-fixed short-extension isomorphism. Conversely $p_{\ast }{i}_{\ast }{H}^{\mathrm{\prime }}=0$ for every $H^{\mathrm{\prime }}$ by $pi=0$, since pushout along the zero map is split. Therefore exactness holds at ${\mathrm{YExt}}^1(X,B)$.

At ${\mathrm{YExt}}^1(X,C)$, [F3] states precisely that $\mathrm{\partial }[F]=0$ if and only if $[F]$ lies in the image of $p_{\ast }$. This proves every asserted exactness in the covariant sequence, with no assertion of surjectivity onto the final term.

The opposite of an abelian category is abelian: kernels and cokernels interchange, biproducts remain biproducts, and the coimage-image isomorphism dualizes. Reversing all arrows therefore identifies its short and two-fold extensions, endpoint-fixed zigzags, pullbacks, pushouts, and Baer operations with the corresponding dual constructions in the original category. Apply steps 1.1–3.1 in that opposite category to $0\to C\to B\to A\to 0$ and $X$. The resulting covariant sequence is exactly the second displayed sequence; its first connecting map pushes $E$ out along $A\to X$, and its last connecting map splices an extension of $A$ by $X$ on the left of $E$. This proves all its asserted exactness.