4 ms·

Topology from the Perspective of Category Theory

42 points by 909832 6y ago

taliesinb 6y ago
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.
- openfuture 6y ago
  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)
  - taliesinb 6y ago
    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?
    - openfuture 6y ago
      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
- neel_k 6y ago
  Do you mean topology, or geometry? For topology, there are two main constructive approaches. The first, better-developed one, can be found in the theory of locales. An easy intro is Vickers' Topology via Logic, from which you can level up to Johnstone's Stone Spaces. This approach is suitable for a background logic which is constructive, but accepts the powerset axiom. If you want to restrict yourself to not just be constructive, but also predicative, then the thing to look at is Sambin's formal topology, which he surveys in his 2001 paper Some Points in Formal Topology. (https://www.math.unipd.it/~sambin/txt/SP.pdf https://www.math.unipd.it/~sambin/txt/SP.pdf)
- m_j_g 6y ago
  more homotopy than topology, but still may be interesitng to you: https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf https://staff.math.su.se/anders.mortberg/papers/cubicalsynth...

42 points by 909832 6y ago

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.

openfuture 6y ago
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)
- taliesinb 6y ago
  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?
  - openfuture 6y ago
    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
neel_k 6y ago
Do you mean topology, or geometry? For topology, there are two main constructive approaches. The first, better-developed one, can be found in the theory of locales. An easy intro is Vickers' Topology via Logic, from which you can level up to Johnstone's Stone Spaces. This approach is suitable for a background logic which is constructive, but accepts the powerset axiom. If you want to restrict yourself to not just be constructive, but also predicative, then the thing to look at is Sambin's formal topology, which he surveys in his 2001 paper Some Points in Formal Topology. (https://www.math.unipd.it/~sambin/txt/SP.pdf https://www.math.unipd.it/~sambin/txt/SP.pdf)
m_j_g 6y ago
more homotopy than topology, but still may be interesitng to you: https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf https://staff.math.su.se/anders.mortberg/papers/cubicalsynth...