The category of Cech closure spaces (a.k.a. pretopological spaces) enables essentially what is described in the Edit/Clarification part of this post. The basic setup is described in the paper

Rieser, Antonio. Cech Closure Spaces: A Unified Framework for Discrete and Continuous Homotopy, Topology and Its Applications, Vol. 296, 2021, https://doi.org/10.1016/j.topol.2021.107613

The construction of a Cech (co)homology functor is an immediate consequence of the results in

Bentley, H. L., Homology and Cohomology for Merotopic and Nearness Spaces, Quaestiones MAthematicae, 6:1-3, 27-47. https://doi.org/10.1016/j.topol.2021.107613

by applying those results to the merotopic structure generated by interior covers of a closure space. (Interior covers are defined below.) However, this is done more explicitly (and, in my opinion, more clearly) for Cech closure spaces in Luis Palacios' 2019 Bachelor's Thesis from the Universidad de Guanajuato, which can be found at https://itarios.github.io/docs/LJPV_MS_Thesis_.pdf

The basic idea is the following. Just as topological spaces have a closure operator which sends a subset $A$ of a topological space $(X,\tau)$ to its topological closure, it's also possible to define useful closure operators which are not idempotent. For example, if one takes a metric space
$(X,d)$ and a positive scale $r>0$, then for any $A\subset X$,

$$c(A) := \{x\in X\mid d(x,A)\leq r \}$$

is a perfectly reasonable closure operation, although the resulting structure is no longer topological (in the sense that $\{X - c(A) \mid A\subset X\}$ is no longer a topology). A function $f:(X,c_X) \to (Y,c_Y)$ between closure spaces is said to be *continuous* (i.e. is a morphism in the category) iff $\forall A\subset X, f(c_X(A)) \subset c_Y(f(A))$. For $r>0$ in the closure space above, this will include many functions which are not topologically continuous. Nonetheless, one can still recover a significant amount of algebraic topology working in such a category.

To construct Cech (co)homology functorially, let $(X,c)$ be a closure space (such as the one constructed from a metric space and a scale above), and define the *interior operator* $i:\mathcal{P}(X) \rightarrow \mathcal{P}(X)$ for any $A\subset X$ by

$$i(A) := X - c(X - A).$$

One now says that a collection of subsets $\mathcal{U}$ of $X$ is an *interior cover* of $X$ iff $X$ is covered by the interiors of the sets in $\mathcal{U}$, i.e.

$$ X = \bigcup_{U\in \mathcal{U}} i(U) $$.

You now run the construction of Cech (co)homology using nerves of interior covers of $(X,c)$ in place of open covers of a topological space, and most of the standard properties of Cech (co)homology can be recovered. In particular, the construction is functorial.

A similar kind of functor may also be constructed for Vietoris-Rips (co)homology using the category of semi-uniform spaces instead of closure spaces, although there are some relationships between these two categories. This is described in

Rieser, Antonio. Vietoris-Rips Homology Theory for Semi-Uniform Spaces. https://arxiv.org/abs/2008.05739

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