Let’s look at an example of a presheaf topos, to see what various things I’ve been talking about actually look like—especially the subobject classifier. Our example will illustrate the connection between topos theory and intuitionistic logic: that is, logic where the law of excluded middle, “p or not p”, fails. Intuitionistic logic goes back to […] […]

Read More Topos Theory (Part 8) — Azimuth

I’m almost done explaining why any presheaf category is an elementary topos, meaning that • it has finite colimits; • it has finite limits; • it’s cartesian closed. • it has a subboject classifier. Last time I explained why such categories are cartesian closed; now let’s talk about the subobject classifier! Subobject classifiers In the […] […]

Read More Topos Theory (Part 7) — Azimuth

I’m explaining why any presheaf category is an elementary topos, meaning that • it has finite colimits; • it has finite limits; • it’s cartesian closed. • it has a subboject classifier. Last time I tackled the first two bullet points; now let’s do the third. For starters, what’s a cartesian closed category, and why […] […]

Read More Topos Theory (Part 6) — Azimuth

An interesting situation has arisen. Some people working on applied category theory have been seeking a ‘killer app’: that is, an application of category theory to practical tasks that would be so compelling it would force the world to admit categories are useful. Meanwhile, the UMAP algorithm, based to some extent on category theory, has […] […]

Read More The Category Theory Behind UMAP — Azimuth

In Part 1, I said how to push sheaves forward along a continuous map. Now let’s see how to pull them back! This will set up a pair of adjoint functors with nice properties, called a ‘geometric morphism’. First recall how we push sheaves forward. I’ll say it more concisely this time. If you have […] […]

Read More Topos Theory (Part 4) — Azimuth

Last time I described two viewpoints on sheaves. In the first, a sheaf on a topological space is a special sort of presheaf Namely, it’s one obeying the ‘sheaf condition’. I explained this condition in Part 1, but here’s a slicker way to say it. Suppose is an open set covered by a collection of […] […]

Read More Topos Theory (Part 3) — Azimuth

Last time I defined sheaves on a topological space this time I’ll say how to get these sheaves from ‘bundles’ over You may or may not have heard of bundles of various kinds, like vector bundles or fiber bundles. If you have, be glad: the bundles I’m talking about now include these as special cases. […] […]

Read More Topos Theory (Part 2) — Azimuth

I’m teaching an introduction to topos theory this quarter, loosely based on Mac Lane and Moerdijk’s Sheaves in Geometry and Logic. I’m teaching one and a half hours each week for 10 weeks, so we probably won’t make it far very through this 629-page book. I may continue for the next quarter, but still, to […] […]

Read More Topos Theory (Part 1) — Azimuth