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April 20, 2007

Cohomology and Computation (Week 20)

Posted by John Baez

This week in our seminar on Cohomology and Computation, we began to see what’s so great about simplices:

  • Week 20 (Apr. 12) - Cohomology and the category of simplices. Simplices as special categories: finite totally ordered sets, which are isomorphic to "ordinals". The algebraist’s category of simplices, Δ alg\Delta_{alg}. Face and degeneracy maps. The functor from Δ alg\Delta_{alg} to Top sending the ordinal nn to the standard (n1)(n-1)-simplex. Simplicial sets. Preview of the cohomology of spaces.

Last week’s notes are here; next week’s notes are here.

A simplex is a special sort of space. A point is a 0-simplex, an interval is a 1-simplex, a triangle is a 2-simplex, a tetrahedron is a 3-simplex, and so on. Here’s a movie of a 4-simplex, rotating in the 4th dimension:

But beneath the level of topology, there’s a deeper level where a simplex is just a finite ordinal. And beneath that, a simplex is a special sort of category! We’ll learn more about the amazing algebraic properties of simplices later on.

Posted at April 20, 2007 5:57 AM UTC

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Read the post Cohomology and Computation (Week 19)
Weblog: The n-Category Café
Excerpt: The origins of cohomology in the study of 'syzygies', or relations between relations.
Tracked: April 20, 2007 8:40 PM
Read the post Cohomology and Computation (Week 21)
Weblog: The n-Category Café
Excerpt: Why mathematicians like to take algebraic gadgets and topological spaces and turn them into simplicial sets.
Tracked: April 22, 2007 2:57 AM

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