Spectral sequence subquotient and local lifting calculus

A cokernel $q$ is epic: if $uq=vq$, uniqueness in its universal property applied to that common composite gives $u=v$. A kernel is monic by the corresponding uniqueness. Thus [F1] factors every $f:A\to B$ as $A\stackrel{e}{\to }\mathrm{im}f\stackrel{m}{\to }B$ with $e$ epic, $m$ monic and $A\mathrm{/}\ker f\cong \mathrm{im}f$.

For $f:A\to X$ and $e:B\to X$, set $D=f{\pi }_A-e{\pi }_B:A\oplus B\to X$ and $P=\ker D$. Its projections $\alpha :P\to A$, $\beta :P\to B$ satisfy $f\alpha =e\beta$. Any pair $(a,b)$ with $fa=eb$ factors uniquely through $\ker D$, proving the pullback property, also when one object is zero.

If $f$ is monic, [F2] and [F3] identify its coimage projection with an isomorphism; the factorization in step 1.1 therefore makes $f$ a kernel of its cokernel. If $f$ is epic, its image inclusion is an isomorphism by the kernel clause of [F3], so $f$ is a cokernel of its kernel. In particular a map both monic and epic is an isomorphism.

If $e$ is epic, $D$ is epic since $D{\iota }_B=-e$. If $q:A\to Q$ satisfies $q\alpha =0$, then $q{\pi }_A$ kills $\ker D$; step 2.1 gives $v:X\to Q$ with $vD=q{\pi }_A$. Composing with ${\iota }_B$ gives $ve=0$, hence $v=0$; composing with ${\iota }_A$ gives $q=0$. Apply this to the difference of any two maps agreeing after $\alpha$: $\alpha$ is epic.

Define $U+V=\mathrm{im}(U\oplus V\to X)$ and $U\cap V=U{\times }_{X}V$. Both have their least-upper-bound and greatest-lower-bound properties: a common upper subobject contains the biproduct map and hence its image by normality, while a common lower subobject factors through the pullback. A map $h:M\to M^{\mathrm{\prime }}$ with $h(N)\subseteq N^{\mathrm{\prime }}$ gives a unique $\bar{h}:M\mathrm{/}N\to M^{\mathrm{\prime }}\mathrm{/}{N}^{\mathrm{\prime }}$ because the composite $M\to M^{\mathrm{\prime }}\to M^{\mathrm{\prime }}\mathrm{/}{N}^{\mathrm{\prime }}$ kills $N$.

For $h:T\to \mathrm{im}f$, pull back the epic $e$ of step 1.1 along $h$. Step 3.1 gives an epic $t:T^{\mathrm{\prime }}\to T$ and a lift $a:T^{\mathrm{\prime }}\to A$ with $ea=ht$. Equalities after $t$ descend by epic cancellation. If $ht$ factors through a subobject $N\hookrightarrow Y$, then $(Y\to Y\mathrm{/}N)ht=0$, hence $(Y\to Y\mathrm{/}N)h=0$, and normality in step 2.1 factors $h$ through $N$. Conversely, a factorization of $h$ gives one of $ht$ by composition. Finite repetitions compose epimorphisms; no section or choice of global representatives is asserted.

The kernel of $U\to X\mathrm{/}V$ is $U\cap V$, since $V=\ker (X\to X\mathrm{/}V)$ by step 2.1. Its image agrees with that of $U+V\to X\mathrm{/}V$: after the epic $U\oplus V\to U+V$ the latter map is $[\bar{u},0]$, which has the same image as $\bar{u}$. The kernel of $U+V\to X\mathrm{/}V$ is $V$, so step 1.1 identifies this image both with $U\mathrm{/}(U\cap V)$ and with $(U+V)\mathrm{/}V$, proving the first isomorphism.

For $V\subseteq U$, step 5.1 makes $U\mathrm{/}V\to X\mathrm{/}V$ monic. Maps $X\mathrm{/}V\to T$ killing $U\mathrm{/}V$ correspond uniquely to maps $X\to T$ killing $U$: each direction follows by composing with the epic quotient from $X$ and then applying cokernel descent. Thus $X\mathrm{/}U$ and $(X\mathrm{/}V)\mathrm{/}(U\mathrm{/}V)$ satisfy the same cokernel property, giving inverse compatible maps and the second isomorphism.

For $U\subseteq W$, let $h:T\to W\cap (U+V)$. Step 4.1 locally writes its composite into $X$ as $u+v$, with $u$ through $U$ and $v$ through $V$. Since both $h$ and $u$ factor through $W$, their difference $v$ factors through $W\cap V$. Thus $h$ locally, and by step 4.1 globally, factors through $U+(W\cap V)$. The reverse inclusion holds because both summands lie in $W$ and in $U+V$. This proves modularity. All comparison maps above are characterized by the original inclusions and quotient projections; composing with a map preserving the subobjects yields the same characterizations. Uniqueness and epic cancellation prove naturality and identity/composition compatibility.