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Stream: learning: questions

Topic: hom is lax monoidal


view this post on Zulip Jules Hedges (Jan 11 2022 at 12:03):

More of a comment than a question: If C\mathcal C is a monoidal category, then hom:Cop×CSet\hom : \mathcal C^\mathrm{op} \times \mathcal C \to \mathbf{Set} is lax monoidal (for the cartesian product on Set\mathbf{Set}), but it is NOT strong monoidal. That is, there is a non-invertible natural transformation hom(x,y)×hom(x,y)hom(xx,yy)\hom (x, y) \times \hom (x', y') \to \hom (x \otimes x', y \otimes y'). It takes about 5 seconds thinking to realise this, but I made exactly this error in one of my papers, and peer review didn't catch it either

view this post on Zulip Jules Hedges (Jan 11 2022 at 12:05):

See also: https://mathoverflow.net/questions/137359/what-are-the-conditions-for-the-hom-functor-to-be-strong-monoidal

view this post on Zulip Paolo Perrone (Jan 11 2022 at 13:18):

The failure of this functor from being strong monoidal could be a good description for the "complexity" of things.

view this post on Zulip David Egolf (Jan 11 2022 at 15:46):

Paolo Perrone said:

The failure of this functor from being strong monoidal could a good description for the "complexity" of things.

It would be really interesting if someone could illustrate this with a small example.

view this post on Zulip Jules Hedges (Jan 11 2022 at 16:06):

For Set\mathbf{Set} with cartesian product, the elements of hom(X×X,Y×Y)\hom (X \times X', Y \times Y') in the image of the laxator are exactly those functions that can be factored as f(x,x)=(f1(x),f2(x))f (x, x') = (f_1 (x), f_2 (x')). So "complexity" in this example is the ability of every output of a function to depend on every input

view this post on Zulip David Egolf (Jan 11 2022 at 16:19):

Jules Hedges said:

For Set\mathbf{Set} with cartesian product, the elements of hom(X×X,Y×Y)\hom (X \times X', Y \times Y') in the image of the laxator are exactly those functions that can be factored as f(x,x)=(f1(x),f2(x))f (x, x') = (f_1 (x), f_2 (x')). So "complexity" in this example is the ability of every output of a function to depend on every input

Interesting! This is similar to a situation in engineering where we say "there is no cross-talk". If we view ff as a message sent across a channel from two sources xx and xx', then the condition f(x,x)=(f1(x),f2(x))f (x, x') = (f_1 (x), f_2 (x')) says that the message from the two sources decomposes into messages generated by each source individually.
So, I suppose the more "complex" case would correspond to when there is cross-talk.

view this post on Zulip John Baez (Jan 12 2022 at 17:05):

This business about the laxator also shows up in the theory of decorated cospans, where we use it to describe, e.g. how there are more chemical reactions involving molecules A,B,C,D,E than those the consist of a reaction involving A,B,C and a reaction involving D,E.