User talk:Lethe/list of categories

ready for article space?

So, ahh, why isn't this in the article space? linas 22:54, 20 July 2006 (UTC)

Well, mostly I haven't moved it into article space because it's not finished. I also have some doubts about its maintainability, and it is noncompliant with certain wikipedia policies in a way which I don't know how to fix. Nevertheless, it's a really cool list. I like reading down the right-hand column, the category-theoretic relations among the various categories describe with abstract nonsense enormous swaths of mathematical theory with only a few words. Category theory is a broad brush. I would like to see this make it to article space some day. -lethe ^{talk +} 23:41, 20 July 2006 (UTC)

Here's an additon then: the Segre embedding is the categorical product on the category of projective varieties (and I assume for all varities as well, then; not enirely clear on that). As you point out, that table is scary to maintain. linas 01:52, 21 July 2006 (UTC)

graphs

I see there's a category of directed graphs (but no article yet). Any clue if there's a category of trees/binary trees? I note that the p-adic numbers, as well as the real numbers when represented as strings of integers, are actually trees (viz yea olde 0.999..=1.000... debate). I also note that grassmanians (and thus supernumbers, supermanifolds and other bits of supersymmetry) can be represented as binary trees (although not uniquely). The various cantor sets are also binary trees, as is alpeh_one = powerset of aleph_zero. linas 23:03, 20 July 2006 (UTC)

Graphs are to categories as sets are to groups. If you take away the multiplication in a group, you have a set, and if you take away the multiplication from a category, then you have a graph. There is of course a forgetful functor and a free functor in this business, and this is what makes the category of graphs worth talking about when doing category theory. As for whether trees form a category, the answer must be yes. A tree is just a directed acyclic graph, right? Maybe you have to say something about roots too. Anyway, what might be an interesting category is this: is there an adjoint functor to the forgetful functor from Tree? That is, can we build a canonical tree from every graph in a coherent way? -lethe ^{talk +} 23:36, 20 July 2006 (UTC)

Thanks, I'll have to digest. The roots of the infinite binary tree that is the Cantor set are, well, those things the real numbers, and I admit that this is whats holding my interest. I have some garbled ideas that I'm trying to sort through. linas 01:52, 21 July 2006 (UTC)

Well, yes, the tree is a directed acyclic graph. As such, it can be represented by a set. (nodes are elements of the set that are also subsets). The point is that that there are many non-trivial ops that can be defined on trees: besides just things like the the rebalancing of trees on computer science (see e.g. red-black tree in comp sci), more generally, *all* of comp sci is isomorphic to lambda calculus which is just a set of operations on finite trees. In the infinite-tree case, such as the cantor set, I'm interested in the "hyperbolic rotations" and bits of surgery on the tree. So, as you see, its already hopelessly general, and so I was looking for some discussion that may have weeded through this generality and fixated on perhaps a more managable set of concepts. linas 02:34, 21 July 2006 (UTC)