Delta Functors and Universality — Examples

Restrict the integer-indexed homology family of [L1] to complexes that vanish in negative degrees. This full subcategory is closed under kernels and cokernels degreewise, hence is abelian, and its short exact sequences have $H_{-1}=0$.

The exactness and naturality axioms required in [L3] are supplied by [L2]; step 1.1 makes the integer-indexed long sequence terminate as $H_0(C)\to 0$. Therefore the restricted family is a homological delta functor indexed by $n\ge 0$.

The displayed homological and cohomological families are exactly the two families identified in [L1].

Therefore both are universal delta functors.

By [L1], the given short exact sequence yields an exact segment $L_n^{P}F(Q)\to L_n^{P}F(A)\stackrel{{\mathrm{\partial }}_n}{\to }{L}_{n-1}^{P}F(K)\to L_{n-1}^{P}F(Q)\mathrm{.}$

Since $Q\in \mathcal{D}$ is projective and $n>1$, [L2] gives $L_n^{P}F(Q)=L_{n-1}^{P}F(Q)=0$. Therefore [L3] turns the connecting map in step 1.1 into the displayed isomorphism.

By [L1], the given short exact sequence yields an exact segment $R_I^{n-1}F(J)\to R_I^{n-1}F(C)\stackrel{{\mathrm{\partial }}^{n-1}}{\to }{R}_I^{n}F(A)\to R_I^{n}F(J)\mathrm{.}$

Since $J\in \mathcal{D}$ is injective and $n>1$, [L2] gives $R_I^{n-1}F(J)=R_I^{n}F(J)=0$. Hence [L3] makes the connecting map in step 1.1 an isomorphism.

The defining equation for $u_1(A)$ is exactly the homological case of [L1] with $n=1$.

Item [L2] removes dependence on the chosen effacement and supplies the required compatibility with connecting morphisms, so the map from step 1.1 is the correct first higher component of the universal extension. This is the degree-one pattern used abstractly in [L3].

Start with the homology delta functor from [L2]. Keep every functor $H_n$ unchanged and keep every connecting map unchanged except on one chosen short exact sequence with nonzero connector from [L3], where replace $\mathrm{\partial }$ by $-\mathrm{\partial }$. Each individual long exact sequence remains exact.

Choose a distinct isomorphic copy of the cone sequence in [L3], and alter the connector on only the original sequence. The chosen isomorphism of short exact sequences induces isomorphisms on the two homology groups. Before the alteration, naturality identifies the two routes around the connecting square with the same map $\pm 1:\mathbb{Z}\to \mathbb{Z}$. After changing exactly one connector to its negative, the two routes are opposite maps $\pm 1$ and $\mp 1$, which are unequal over $\mathbb{Z}$. Hence this naturality square fails, and [L1] shows that the altered family is not a homological delta functor.

By [L1], both $(L_n^{P}F)$ and $(L_n^{Q}F)$ are universal homological delta functors, and [L3] supplies their natural degree-zero identifications with $F$.

Choose the natural degree-zero identifications with $F$ from step 1.1. Applying [L2] yields the unique isomorphism of delta functors whose degree-zero part corresponds under those identifications to ${\mathrm{id}}_F$; its higher components are then forced.