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Avi Craimer (Apr 07 2021 at 14:01):Hi,
I'm defining an algebraic structure in terms of a set some functions on that set, and some added constraints that must hold for those functions. Is there a way to translate this into a functor from some index category into Set?
In my example, I start by defining the category of free monoids with 3 generators by taking an index category that has three freely composing non-identity arrows, and taking all functors from this category into Set. However, I want to get a sub-category of this, which has a few additional constrains. For example, if we call the endofunctions f, g, and h, I want to say that if an element is fixed under f and g i.e., if f(x) = g(x) = x , then h(x) must be a special element p. In addition, I want to say that there are three distinct special elements {a,b,c} in the monoid's underlying set, which are such that for all elements x, h(x) is not an element of {a,b,c}.
I know there is probably some super general/abstract way to do this in general using Yoneda's lemma and such, but I'm wondering if there is a more concrete way to think about this for simple definitions like the example I gave.
Joe Moeller (Apr 07 2021 at 14:04):Perhaps Lawvere theories are what you're looking for?
Avi Craimer (Apr 07 2021 at 14:09):Joe Moeller said:
Perhaps Lawvere theories are what you're looking for?
Do you know of an accessible introduction to this?
Nathanael Arkor (Apr 07 2021 at 14:21):Algebraic Theories: A Categorical Introduction to General Algebra is very good.
Fawzi Hreiki (Apr 07 2021 at 14:28):If you want something slightly easier and which also deals with the monad approach to universal algebra, you could take a look at Manes' book 'Algebraic Theories'
Morgan Rogers (he/him) (Apr 08 2021 at 10:21):Avi Craimer said:
In my example, I start by defining the category of free monoids with 3 generators by taking an index category that has three freely composing non-identity arrows, and taking all functors from this category into Set.
This isn't a category of free monoids..! It's the category of left actions of the free monoid on 3 generators.
Avi Craimer (Apr 08 2021 at 14:59):@_Morgan Rogers (he/him)|277473 said:
This isn't a category of free monoids..! It's the category of left actions of the free monoid on 3 generators.
Thanks for this clarification. I know it's not the category of free monoids, but I thought it was the category of free monoids with three generators. Since you can compose the morphisms in either direction, I don't understand the "left actions" part.
Nathanael Arkor (Apr 08 2021 at 15:18):A one-object category is a monoid. So if you freely generate a one-object category from 3 arrows, you get the free monoid on 3 generators. Taking presheaves on a monoid gives you the left-actions on that monoid.
John Baez (Apr 08 2021 at 15:31):A left action of a monoid M, by the way, is a set S together with a map f(m): S S for each m M such that f(mm') = f(m)f(m') and f(1) = . So, it's a functor from that monoid to Set.
(It's a presheaf on the opposite of that monoid. But a free monoid is isomorphic to its opposite.)
John Baez (Apr 08 2021 at 15:32):So yeah, @Avi Craimer described the category of left actions of the free monoid with 3 generators.
Morgan Rogers (he/him) (Apr 09 2021 at 07:55):The point being that there is no multiplication operation on the set being acted upon, only operations corresponding to the element of the free monoid.