Let $\mathbf{C}$ be a category in which we have some notion of (co)homology (say, groups with the usual cohomology, posets with the homology of the geometric realisation). We say that an embedding $F: \mathbf{C} \to \mathbf{Cat}$ of $\mathbf{C}$ preserves (co)homology if, for each object $X$ of $\mathbf{C}$, the (co)homology of $X$ agrees with the (co)homology of $F(X)$ (defined as the higher (co)limits of a suitable (that is: "naturally" derived from what we're taking our coefficients in over on $\mathbf{C}$) sheaf on $F(X)$).

For example, the usual functor $F_1$ embedding the category $\mathbf{Grp}$ of all groups into $\mathbf{Cat}$ by treating a group $G$ as a category with one object $\ast$ and $(\mathrm{Hom}(\ast,\ast),\circ) \cong G$ preserves cohomology: for a group $G$, the $n$th cohomology of $G$ with coefficients in a $G$-module $M$ is exactly the $n$-th higher limit of the presheaf on $F_1(G)$ taking the value the underlying abelian group of $M$ at $\ast$ and sending the morphism corresponding to each $g \in G$ to the endomorphism given by the action of $g$ on $M$.

Similarly, the embedding $F_2$ of the category of posets into $\mathbf{Cat}$ sending a poset $P$ to the category whose objects are the elements of $P$ with exactly one morphism $x \to y$ if $x \leq y$ in $P$, and no morphisms otherwise preserves cohomology, for essentially trivial reasons.

So, that brings us to my question: which other cohomology-preserving functors from "interesting" categories exist? In particular, can we do this for CW-complexes (or some other reasonable subcategory of $\mathbf{Top}$)?

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    $\begingroup$ For topological spaces, one can take the category of simplices of the singular simplicial set. $\endgroup$ – Dmitri Pavlov Mar 8 at 22:47

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