Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I have ideas about applying magnitude (a la Leinster and Meckes) to at least one real-world problem and can't help noticing magnitude homology of graphs.

Generally, I deal with finite/discrete structures a lot and so would like to see what hammers are out there so as to keep my eyes peeled for any matching nails. That is,

I am hoping to get an exhaustive picture of what topological theories have been devised for "nice" finite objects, with an eye towards investigating any that have (or at least admit) practically computable algorithms.

Because of the algorithmic focus, I am not sure if I should include, e.g., any Khovanov or Kontsevich stuff, which never seemed very accessible and/or applicable to my naive eye, and I certainly don't understand it.

Google Scholar could answer this in principle, but an hour or so has convinced me that this is a worse idea than asking here.

To avoid degeneracy, I will omit the algebraic topology of finite topological spaces a la McCord. My first stab (which I'm fairly confident would omit something that I've already forgotten, let alone never heard of) is:

• Abstract simplicial complexes (included only for the sake of completeness)
  • simplicial (co)homology
  • discrete Morse theory
• Posets
  • (co)homology of the associated abstract simplicial complex
• Relations (and/or bipartite graphs)
  • Dowker homology = simplicial homology of an associated abstract simplicial complex
    • (eminently computable)
• Metric spaces
  • homology theory of Barcelo et al.
  • magnitude homology
    • (note that this generalizes Hepworth-Willerton graph homology, so tenatively not including that under "graphs")
• Graphs
  • degenerately, simplicial stuff
• Digraphs
  • path (co)homology and homotopy theories of Grigor'yan et al.
    • (homotopy theory of Barcelo et al. apparently equivalent to this)
    • (homology computable at reasonable scale, but with exponential complexity in dimension)

For some of these (e.g., magnitude homology) I don't yet have a real idea of the computational practicality, and I would be especially interested in any observations in this vein and/or pointers to code.

I would encourage answerers to update this list as well as answering below.