Cech cohomology - what is it really?

I insist vehemently that I am a logician. My colleagues tend to believe me less and less each time we speak, because every time they see me, I seem to be carrying around a book on algebraic topology, homological algebra, or differential forms. It is true that I have not spent as much time as I should lately on logic proper. But my interest as a logician led me to topos theory, and the deeper I started to get into topos theory, the more I enjoyed it and wanted to appreciate it on its own standing - not merely as a setting for the semantics of intuitionistic logics, but as a rich branch of mathematics, invented to provide a setting for Weil's hypothesized cohomology theory, and drawing inspiration and motivation from all over algebraic topology and algebraic geometry. Since then I have begun to read broadly into classical geometry, trying to understand the historical context in which topos theory was created, and answer for myself the question "What is cohomology, really?" This seems to be sufficiently broad (and impossible to answer) that I'm not sure I'll ever understand it. But certainly I have made progress. The first time I saw the definition of Cech cohomology in Hartshorne, about a year ago, I was completely at a loss. Today, I think I've learned enough about Cech cohomology to say that I confidently understand what it is trying to do, relative to sheaf cohomology - maybe not in some insane higher-categorical sense, I'm sure Urs Schreiber or someone like this will always be around to say "well, really this is nothing but the study of  (complete gibberish)" - but I have found an interpretation with which I am satisfied. Cech cohomology is the straightforward importation of sheaf cohomology into the realm of simplicial sets.

Attached:
A short comment I've written on Cech cohomology:
https://www.math.wisc.edu/~nicodemus/Cech_Cohomology_and_Cech_Methods.pdf
https://www.math.wisc.edu/~nicodemus/Cech_Cohomology_and_Cech_Methods.tex

"The purpose of this note is to establish some basic facts dealing with the relationship between the geometric realization of a simplicial complex, the nerve associated to an open cover of a space, and partitions of unity subordinate to an open cover. The contents are not difficult: they are just an elaboration on some straightforward exercises in the chapter of Spanier’s Algebraic Topology textbook on simplicial complexes. Towards the end, we prove a useful theorem: if \(X\) is a compact Hausdorff space, and \(K\) is a simplicial complex, then homotopy classes of maps from \(X\) into \( | K |\) can be approximated by taking an open cover \(\mathcal{U}\) of \(X\) and looking at contiguity classes of maps from the nerve of \(\mathcal{U}\) into K, where a contiguity class is a purely combinatorial analogy of homotopy class defined for maps between simplicial complexes. Taking the colimit over locally finite open covers \(\mathcal{U}\), the approximation becomes perfect. We try to establish the intuition that the nerve of a good open cover of \(X\) is a good approximation to a triangulation of the space \(X\). Notably absent is the theorem that if the open cover \(U\) has the property that all finite intersections \(U_{i_1}\cap \dots\cap U_{i_n}\) are contractible, then \(X\) is homotopy equivalent to \( | K(\mathcal{U}) | \). Once I learn the proof of this (if ever!) I hope to come back later and add it."
https://www.math.wisc.edu/~nicodemus/Partitions_of_Unity_and_the_Nerve.pdf
https://www.math.wisc.edu/~nicodemus/Partitions_of_Unity_and_the_Nerve.tex
 

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