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Stream: learning: questions

Topic: Monoid object associativity

Shea Levy (Oct 20 2020 at 19:38):

Why is the associativity law for monoid objects $\mu \circ (1 \otimes \mu) \circ \alpha \equiv \mu \circ (\mu \otimes 1)$ instead of just $(\mu \otimes 1) \equiv (1 \otimes \mu) \circ \alpha$ ?

James Wood (Oct 20 2020 at 19:44):

These laws are probably easier to understand if you ignore the associators and unitors of the monoidal category (as justified by the coherence theorem for monoidal categories). In this sort of stuff, it's common to switch to string diagrams, which, among other things, are a way to ignore associators and unitors.

James Wood (Oct 20 2020 at 19:46):

I think in Set, a counterexample to the latter is to take the (ℕ, 0, +) monoid and some input like ((1, 2), 3).

Shea Levy (Oct 20 2020 at 19:47):

Ah right I was confusing the tensor for the monoid product. Thanks!

James Wood (Oct 20 2020 at 19:47):

Then, the LHS results in (3, 3), whereas the RHS results in (1, 5) (if I got the types right).

James Wood (Oct 20 2020 at 19:48):

And fortunately, one more addition makes these match up!

Shea Levy (Oct 20 2020 at 19:49):

James Wood said:

These laws are probably easier to understand if you ignore the associators and unitors of the monoidal category (as justified by the coherence theorem for monoidal categories). In this sort of stuff, it's common to switch to string diagrams, which, among other things, are a way to ignore associators and unitors.

Any recommended resource for getting started with string diagrams?

James Wood (Oct 20 2020 at 19:55):

I'm sure others have actual recommendations, but I don't. I just picked them up from people around me plus being similar enough to circuit diagrams that intuition carries over.

John Baez (Oct 20 2020 at 20:30):

Some introductions to string diagrams include:

• Bob Coecke and Aleks Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning.

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta stone.

• Peter Selinger, A survey of graphical languages for monoidal categories.

I think these are in order from easy to hard. String diagrams are so simple that they could easily be taught in elementary school - that's why Bob Coecke wrote the paper Kindergarten quantum mechanics. You don't need to know anything about category theory or quantum mechanics to understand them, though they do make category theory and quantum mechanics easier to understand.

John Baez (Oct 20 2020 at 20:37):

Because string diagrams are applicable to so many things, introductions to them tend to talk about these things - which may make string diagrams seem harder to understand than they really are!

Henry Story (Oct 20 2020 at 22:40):

As an example of String diagrams applied to a particular field, I found Knowledge Representation in Bicategories of relations by @Evan Patterson to be very helpful. It is a good introduction to very simple string diagrams for those who know RDF, and for those who know string diagrams a good intro to RDF. (knowledge seems to flow two ways)

John Baez (Oct 21 2020 at 04:03):

I don't even know what "RDF" means. :cry:

John Baez (Oct 21 2020 at 04:05):

I think most string diagram intros are somewhat subject-specific - "String diagrams and X".

Henry Story (Oct 21 2020 at 08:06):

RDF is the Resource Description Framework. A logic for the Web developed at the W3C. One can think of it as a Regular logic (and extend it to a geometric logic as per Evan's paper). The main difference with normal first order logics is that it uses URLs as identifiers, which allows people to publish their descriptions on their web servers and link to descriptions on other servers. Ie a hyperlinked logic framework.

An important use case is the creation of decentralised secure social networks, where each of us could publish their content at home without the need for centralised services we have now.

John Baez (Oct 21 2020 at 15:38):

Thanks! @Brendan Fong and @David Spivak have a nice (rather high-powered) paper on a graphical calculus for regular logic, so hopefully that's connected to @Evan Patterson's paper.

Henry Story (Oct 21 2020 at 17:34):

yes, it's even referenced in there :smile:
Patterson in "Bicategories of Relations" also references the Functorial DB work by Spivak, who 10 years ago pointed out that a functor from a small category (thought of as schemas) to Set is a DataBase instance, and the Grothendieck construction of that functor is equivalent to (the structure of) RDF. Patterson adapts this to a bicategories of relations: Functors from bicategories of relations to Rel and the Grothendieck construction of those functors appear there too. This matches more directly RDF since that is relational. The two are inter-definable though.