
Topology from the Perspective of Category Theory - 909832
https://topology.mitpress.mit.edu/
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taliesinb
Can anyone recommend a finitist, constructivist approach to topology? I’m
really most interested in discrete geometry, but a geometry that doesn’t try
to preserve traditional concepts like curvature, angle, volume etc, but builds
them up from scratch from finite graphs. A discrete-first approach, like how
Lattice Boltzmann methods are approximated by Navier Stokes rather the way FEM
solvers discretize Navier Stokes.

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openfuture
The requirement for finiteness means you are just working with lattices.
Topology approached via frame homomorphisms (like in the nlab book) makes this
clear.

Disclaimer: not a mathematician (yet)

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taliesinb
Thanks! By nlab book you mean the whole nlab project? Also, can you help me
understand the connection to lattices? Do they model intersection and union of
open sets? Why is finiteness needed for lattices to be appropriate?

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openfuture
I meant [1] they use frames [2] to explain what a topology is but in a finite
setting the frame becomes a lattice because you only have finite meets and
joins.

[1]:
[https://ncatlab.org/nlab/show/Introduction+to+Topology](https://ncatlab.org/nlab/show/Introduction+to+Topology)
[2]:
[https://ncatlab.org/nlab/show/frame](https://ncatlab.org/nlab/show/frame)

