Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?

Question

Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\underline{\mathbb{Z}},-)$ and so $$H^i(X,-)=\operatorname{Ext}^i(\underline{\mathbb{Z}},-)=\hom_{\mathsf{D}(X)}(\underline{\mathbb{Z}},-[i]).$$ We can then use the adjunction to write this as $H^i(X,-)=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*- [i])$. In particular, we have that $H^i(X,\underline{\mathbb{Z}})=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*p^*\underline{\mathbb{Z}} [i])$. The same kind of reasoning shows that $H^i_c(X,\underline{\mathbb{Z}})=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_!p^*\underline{\mathbb{Z}} [i])$.

Now, most references on Verdier duality would call $\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*p^!\underline{\mathbb{Z}} [-i])$ the $i$-th homology of $X$. I get that this is defined precisely in a way that recovers Poincaré duality in its general form. Is there more to it? Why does this deserve to be called a homology group?

Answer 1

One nice geometric way to view homology is a measurement of your space $X$ given by probing $X$ with other, nicer spaces, for instance, singular homology probes with simplices. One good reason to call this abstract thing homology is that it too admits elements corresponding to nice enough maps from test objects into $X$.

Lets work in a general six functor formalism, but I'm secretly thinking of constructible sheaves on nice spaces. Then we have our unit object $\mathbf{1}$ ($\underline{\mathbb{Z}}$ in our case) and a dualising object $p^!\mathbf{1}:=\omega$, in our case, this is the usual dualising sheaf $p^!\underline{\mathbb{Z}}$.

Then we can define a "smooth" object $Z$ to be one which admits an isomorphism $\gamma:\mathbf{1}_Z\rightarrow \omega_Z[-d_Z]$, for some integer $d_Z$. In the constructible/sheafy setting, manifolds give plenty of examples of "smooth" objects, and $d_Z$ is the dimension.

Then if we have $Z$ a closed subset of $X$, or more generally, if $i_*\cong i_!$ for $i:Z\rightarrow X$, then we have the following map, where all arrows are coming from adjunctions in our formalism, except the second one which says $Z$ is "smooth".

$$\mathbf{1}_X\rightarrow i_*\mathbf{1_Z}\xrightarrow{i_*\gamma} i_*\omega_Z[-d_Z]\xrightarrow{\cong}i_!\omega_Z[-d_Z]\rightarrow\omega_X[-d_Z]$$

So the formalism gives us elements of this group for each "smooth" closed subset $Z$ in $X$, so in my mind, this deserves the label of homology, it admits a direct link to the geometry of nice subsets of your space!