Applied Category Theory. Chapter 4, lecture 1 (Spivak)

Applied Category Theory
MIT Course 18.S097
Independent Activities Period (IAP) 2019
Taught by David Spivak and Brendan Fong
© Spifong

That’s a great description that makes the book click for me: “Back up in X or move forward in Y, the yesses remain yesses.”

Effectively this is a way of dynamic programming?

[Kindar]

00:00  Collaborative design.
03:26  Antichains in the weight × velocity partial order.
06:46  Profunctors and feasibility relations.
12:50  An opposite of a category.
14:02  Profunctors.
17:20  Quantales.
19:56  Free discussion.
23:48  A recapitulation of enriched categories. (Refer to chapter 2, lecture 2.)
25:35  Hasse diagrams.
27:14  A collage of a boolean profunctor.
32:50  A functor is a profunctor.
36:06  Reading total cost from a cost enriched profunctor.
37:41  A monotone map is a profunctor.
41:47  Composition of profunctors.
46:12  What is the next lecture going to be about.
46:43  Algorithmic determination of optimal antichains.

[phreedeisele]

anti-chain : given three points A (2, 8) and two candidates B (3, 4) and C (4, 6); A-B and A-C each forming monotonically decreasing fragments. But B-C does not form such a fragment. Does A-B belong to an anti-chain? or is there an additional rule?

from wikipedia: “an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.”
This results in a monotone set but being such a set is not a sufficient condition.

[nathanielvirgo]

is it really true that a functor is a kind of profunctor? In general it seems not to be possible to have a profunctor where every object "has exactly one place to go, " because if there's a "heteromorphism" (an arrow in the collage) from A in C to X in D, and also there's a morphism from X to Y in D, then there has to be a heteromorphism from A to Y, contradicting the "exactly one" condition. Because of this, I always thought that functors and profunctors are just different things.