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Stream: event: ACT20

Topic: Tutorial: Monads and Comonads (Paolo Perrone)

Paolo Perrone (Jun 16 2020 at 14:02):

Hi all! This is the discussion thread for my tutorial, "Monads and comonads".

Paolo Perrone (Jun 16 2020 at 14:03):

Abstract:
We'll introduce monads and comonads, and give some intuition for how they often appear in applications. We'll then talk about Kleisli and Co-Kleisli categories, and show how you can use them to model contexts with either extra outcomes, or extra information.

Paolo Perrone (Jul 02 2020 at 02:45):

When and where: Sunday July 5, 20:00 UTC (4 pm EDT)
Zoom meeting: https://mit.zoom.us/j/7055345747
See also the main website.

Paolo Perrone (Jul 03 2020 at 15:46):

If anyone would like to do some pre-reading, most of the material I will present is taken from my "Notes in Category Theory with examples from basic mathematics", https://arxiv.org/abs/1912.10642, Chapter 5.

Paolo Perrone (Jul 03 2020 at 15:48):

By the way, Christian Williams will be there too, teaching with me!

Paolo Perrone (Jul 05 2020 at 19:56):

We are about to start!

Deepak Vaid (Jul 05 2020 at 20:12):

Paolo Perrone said:

We are about to start!

Unfortunately, its 1:40 am here so I will have to skip the live session. I will catch up with the recorded version though.

Paolo Perrone (Jul 05 2020 at 22:13):

For the ones who asked about probability monads: you can find many explanations here: https://www.youtube.com/playlist?list=PLaILTSnVfqtIebAXFOcee9MvAyBwhIMyr as well as here: https://www.youtube.com/playlist?list=PLSx1kJDjrLRSKKHj4zetTZ45pVnGCRN80

Christian Williams (Jul 06 2020 at 02:13):

If you are interested in monads for (universal) algebra, see https://ncatlab.org/nlab/show/Lawvere+theory and the references therein, such as https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf.

Here's a good reference on algebraic theories: https://web.math.rochester.edu/people/faculty/doug/otherpapers/Adamek-et-al-book.pdf

Here's the entry in Bartosz Milewski's Category Theory for Programmers: https://bartoszmilewski.com/2017/08/26/lawvere-theories/

The nLab page for monads https://ncatlab.org/nlab/show/monad+%28in+computer+science%29 also has a lot of good references.

Christian Williams (Jul 06 2020 at 02:13):

An algebra of $T$, a set $A$ equipped with a function $\alpha:T(A)\to A$, is a set "equipped with $T$-structure": the set $T(A)$ consists of "formal expressions" with variables in $A$, and the map $\alpha$ evaluates these expressions to values. For example if $T$ is the "free monoid" monad, an algebra maps $[a_1,a_2,\dots,a_n]\mapsto a_1a_2\dots a_n$, and the algebra laws amount to saying that this operation is associative and unital, i.e. a monoid.

Christian Williams (Jul 06 2020 at 02:14):

A Lawvere theory $L_T$ is the "category of operations" of a monad $T$; there is an equivalence between finitary monads on $\mathrm{Set}$ and Lawvere theories (see Sec. $1$ of https://arxiv.org/pdf/1112.3076.pdf). A model of $L_T$ in $\mathrm{Set}$ is a functor $M:L_T\to \mathrm{Set}$ which preserves finite products. This maps each "abstract operation" $f:n\to 1$ in $L_T$ to a "concrete operation" $M(f):A^n\to A$.

Christian Williams (Jul 06 2020 at 02:14):

Models of $L_T$ are equivalent to algebras of $T$, and there are advantages to each perspective. Theories are nice because they are categories, where you can draw out each operation and specify equations as commutative diagrams, which many may find to be more explicit and intuitive than an endofunctor on $\mathrm{Set}$ which somehow encapsulates all operations at once. On the other hand, monads are much more general -- as we mentioned briefly at the end, monads exist in any 2-category, and rather than only being maps which add structure to objects, they can themselves be structures of interest.

Christian Williams (Jul 06 2020 at 02:14):

So, both are good! It's pretty amazing how much we've generalized algebra in the past century.

Paolo Perrone (Jul 06 2020 at 08:41):

Hello all! Here's the video.
https://www.youtube.com/watch?v=ryMkvAOJk20&list=PLCOXjXDLt3pYPE63bVbsVfA41_wa3sZOh