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Stream: deprecated: thermodynamics

Topic: operads from operad algebra

Owen Lynch (Dec 31 2021 at 02:56):

Given an operad $\mathcal{O}$, and an operad algebra $\mathrm{F}$, is there a way of constructing a new operad $\mathcal{O}_{\mathrm{F}}$ where the operations include operadic compositions with elements of $\mathrm{F}$?

Owen Lynch (Dec 31 2021 at 02:57):

I.e., suppose I have an operad of linear circuit elements, and an operad algebra of non-linear circuit elements over this operad.

Owen Lynch (Dec 31 2021 at 02:57):

Can I then make an operad of nonlinear circuit elements?

Owen Lynch (Dec 31 2021 at 02:58):

Basically, this would just be adjoining to our original operad a nullary operation for every element of the operad algebra

Owen Lynch (Dec 31 2021 at 02:58):

OK, this seems kind of obvious

Owen Lynch (Dec 31 2021 at 02:59):

This is kind of neat, though, this allows us to talk about diagrams where some of the boxes are filled and some of the boxes are empty

John Baez (Dec 31 2021 at 21:39):

Owen Lynch said:

Given an operad $\mathcal{O}$, and an operad algebra $\mathrm{F}$, is there a way of constructing a new operad $\mathcal{O}_{\mathrm{F}}$ where the operations include operadic compositions with elements of $\mathrm{F}$?

Maybe start with a baby case: say you have a monoid acting on a set. Can you get a new monoid from this, or maybe an operad?

(An operad with only one type and only unary operations is a monoid, and an algebra of this sort of operad is called an "action" of a monoid on a set.)

John Baez (Dec 31 2021 at 21:40):

Oh, maybe you answered my question already - you don't get a new monoid from this, because your new operad has lots of nullary operations.