Category Groups
Apr. 27th, 2019 12:21 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI tend to be slow about series of mathematical posts. Usually, it's because I can't decide how much detail to include. This time around, I've decided to get into the weeds; I can't describe what happened and why it threw me without a mess of detail. You Have Been Warned.
The level of abstraction I'm using in Taxonomy III involves things known as "group algebras". The concept of a group algebra first appeared in 1905, but it's a very classical idea (where "classical" means "before the category theorists came along"). I have no doubt that the great late-19th century algebraists like Cayley, Sylvester, and Grassmann would have had no trouble with it; it's a natural way to talk about symmetry operations on vector spaces. (In my work, the set of, say, all pentagons has the structure of a vector space.) In the past month or so, I've constructed several pieces of machinery involving a particular group algebra, which make it easier to discuss the kinds of polygon-transformations I'm using.
I have to say a few more words about categories. A category is composed of analogues of sets ("objects") and of functions ("arrows"). Every arrow is from one object to another, just as a function sends members of one set to another, and arrows can be composed, if the target of one is the source of another. Sets and functions, vector spaces and linear transformations, topological spaces and continuous functions, vertices of a graph and paths between vertices... Every other construction in category theory is built from those concepts. Generalizing ideas from sets and functions takes some tweaking, usually. For example, the usual definition of a "one to one function" says that a function f is one to one if, whenever x and y are two elements, f(x) and f(y) are the same if and only if x and y are the same. That definition involves elements, which are not a category-theoretic concept. However, it can be rephrased: a function f from A to B is one to one if, whenever g and h are two functions from C to A, the compositions fg and fh are equal if and only if g and h are equal - and that definition can be used in any category.
Now, one odd thing that category theorists do involves duality: if you have a category-theoretic concept, you can try reversing all of the arrows and compositions, and get a new concept. For example, the dual of "one to one function" is this: a function f from A to B with the property that, if g and h are two functions from B to C, then gf and hf are equal if and only if g and h are equal - which turns out to mean the function is "onto". Pretty much any category-theoretic concept can be dualized, and the result is usually indicated by prefixing "co-" to the original name. The Cartesian product of two sets A and B can be characterized by the two projection functions (x,y)->x and (x,y)->y, which jointly have a certain special property. So, the "coproduct" of A and B is described by functions *from* A and B to the coproduct, which functions have a certain special property.
OK. So the category theorists took the concept of an "algebra" - a vector space with a well-behaved multiplication - and dualized it, producing a "coalgebra" or "cogebra". An algebra which is also a cogebra is a "bigebra". These are seriously weird concepts; I have an intuitive feel for how algebras work, but cogebras and bigebras I do not grok.
So, a couple of days ago, I was looking at online articles on group algebras; it's been a while since I worked with them, and I wanted to reassure myself as to just how they work. The article I was reading mentioned that every group algebra is a bigebra, and pointed out how that additional structure worked.
The machinery I'd been devising for Taxonomy III? It's part of the bigebra structure of a group algebra.
This I did not expect. I haven't decided whether to mention that in the writeup.
The level of abstraction I'm using in Taxonomy III involves things known as "group algebras". The concept of a group algebra first appeared in 1905, but it's a very classical idea (where "classical" means "before the category theorists came along"). I have no doubt that the great late-19th century algebraists like Cayley, Sylvester, and Grassmann would have had no trouble with it; it's a natural way to talk about symmetry operations on vector spaces. (In my work, the set of, say, all pentagons has the structure of a vector space.) In the past month or so, I've constructed several pieces of machinery involving a particular group algebra, which make it easier to discuss the kinds of polygon-transformations I'm using.
I have to say a few more words about categories. A category is composed of analogues of sets ("objects") and of functions ("arrows"). Every arrow is from one object to another, just as a function sends members of one set to another, and arrows can be composed, if the target of one is the source of another. Sets and functions, vector spaces and linear transformations, topological spaces and continuous functions, vertices of a graph and paths between vertices... Every other construction in category theory is built from those concepts. Generalizing ideas from sets and functions takes some tweaking, usually. For example, the usual definition of a "one to one function" says that a function f is one to one if, whenever x and y are two elements, f(x) and f(y) are the same if and only if x and y are the same. That definition involves elements, which are not a category-theoretic concept. However, it can be rephrased: a function f from A to B is one to one if, whenever g and h are two functions from C to A, the compositions fg and fh are equal if and only if g and h are equal - and that definition can be used in any category.
Now, one odd thing that category theorists do involves duality: if you have a category-theoretic concept, you can try reversing all of the arrows and compositions, and get a new concept. For example, the dual of "one to one function" is this: a function f from A to B with the property that, if g and h are two functions from B to C, then gf and hf are equal if and only if g and h are equal - which turns out to mean the function is "onto". Pretty much any category-theoretic concept can be dualized, and the result is usually indicated by prefixing "co-" to the original name. The Cartesian product of two sets A and B can be characterized by the two projection functions (x,y)->x and (x,y)->y, which jointly have a certain special property. So, the "coproduct" of A and B is described by functions *from* A and B to the coproduct, which functions have a certain special property.
OK. So the category theorists took the concept of an "algebra" - a vector space with a well-behaved multiplication - and dualized it, producing a "coalgebra" or "cogebra". An algebra which is also a cogebra is a "bigebra". These are seriously weird concepts; I have an intuitive feel for how algebras work, but cogebras and bigebras I do not grok.
So, a couple of days ago, I was looking at online articles on group algebras; it's been a while since I worked with them, and I wanted to reassure myself as to just how they work. The article I was reading mentioned that every group algebra is a bigebra, and pointed out how that additional structure worked.
The machinery I'd been devising for Taxonomy III? It's part of the bigebra structure of a group algebra.
This I did not expect. I haven't decided whether to mention that in the writeup.
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