Embeddings into $\mathbf{Cat}$ preserving cohomology

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 $$\mathbf{Top}$$)?

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