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**Just Chat! / Re: Is Category Theory the well of mathematical philosophy**

« **on:**01/03/2024 06:45:03 »

Mathematicians will always be keen to identify what their most fundamental mathematical objects must be. Physicist's are keen to know what their most fundamental physical object or "atoms" would be, mathematicians are much the same.Well ok. It does seem to be the case that Category Theory has been hailed a success. It isn't "there" yet, likely since nobody knows where "there" is. But I would say it's at least as successful as modern Information Theory, with it's languages and finite state machines to go with. That is, information entropy is also an algebraic entity, no surprise there really because information is not a continuous field. Ahem.

The continuous part of IT is about communicating the stuff, and the Categorists want IT to be understood as part of their lexicon. I don't see a problem. Someone tell me there isn't a problem (just kidding)

It is simple and Mathematicians are probably only working with and manipulating a very small class or category of structures, all built from what are probably just a small set of atomic objects.Indeedly. I wrote down a really nice way to construct a tensor product of group algebras, the groups are the symmetric groups namely S

_{2}, and S

_{1}. This is the trivial case, so you start down the path of induction into larger symmetric groups. There is a vector space that uses an S

_{3}"distinguished" basis, i.e. you have the blindingly obvious (i.e. simple) {1,2}x{1} = {1,2,3}, a "set product", this is also pretty nice. Some permutation diagrams since I want the basis {1,2,3} and Perm{1,2,3}, the set of permutations.

All this is succinctly represented in some diagrams, with vertical and horizontal composition corresponding to inner "vector" and tensor multiplication, resp. with restrictions on the sets. You also invoke the categories Ind and Res as cocategories, it's like a one-page construction of a bunch of highly simplified mathematical thingies.

So yeah, let the good times roll

p.s. I didn't mention that it's been like the way I've kept my keyboard and guitar skills, through self learning. I'm self learning "higher" algebra, and discovering it isn't so hard. It is though, a bit like telling yourself to ignore that voice that says you won't do it.

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