For someone who needs video games explained in terms of moths and "blue shiny things" that was great. Does every island have bridge to every other island, or do you sometimes have to go via one or more intermediary islands?
Please, try to answer in terms of islands and such? I won'r understand technical stuff. Also, a monkey somewhere in the theory (to go with the palm tree) would be fun.
Very many of the islands are connected via at least one bridge to one very specific island - mathematically, very many interesting categories have a forgetful functor to Set - and so you might be able to go between all those by bouncing through this the bridge main central.
Then again, some of the islands are extremely weird. I don't know to what extent bridges always exist but I'd expect some of the bridges to be very rickety and unreliable, requiring you to leave your entire load behind so as to not raze that bridge.
I like the "rickety bridges" idea! In almost all cases, there are very rickety bridges (mathematically, constant functors) from one island to another, but they're mostly of interest to bridge geeks. Of course, if you're coming from a small enough island, then you won't have much stuff to carry with you, and a rickety bridge will do. Bridges which can carry heavy traffic are rarer, though heavily-populated islands (in particular Set
, as michiexile
observes) tend to be served by many large bridges.
There are some curiosities: for instance, there's one (mostly barren - perhaps empty apart from a monkey and a pine tree?) island which has a bridge that goes there from every other island, but no bridges that leave it (bridges are one-way: I should have mentioned that, sorry). There's also an (entirely barren) island which has a bridge going to every other island, and none arriving :-)
I now have a vision of bridges composed of escalators and travelators (occasionally made of wood with big holes between the slats). And I think I actually have a pretty good idea of what you're doing now.
You're a mathematical civil engineer!
I liked that explanation a lot... it even made sense to me at the end of my shift! :)
it's the weekend
no in depth thinking allowed
oh so thats what your doing...nods in understanding...gives up...no sorry angel still don't get it.
You know some bridges have really pretty nuts and bolts and shaped metalwork? He looks at them, and how they all fit together, and while he's on the bridge he looks at the funny people who are scared of heights and stay on the land.
That's more or less it, yes :-)
I understand the metaphor, I just haven't got a clue what the metaphor is for.
There are other types of maths, and his maths is the way the other types lead in to each other?
That really is an awesome metaphor. I can totally appreciate it even if we haven't gotten to the formal definition of a category yet in my Homological Algebra class. :)
Glad you liked it :-)
Without wishing to undermine your course too much, the definition of a category is actually really simple: a category is a directed graph, with an operation called composition on chains of arrows. So a chain of arrows a1 ->^f a2 ->^g a3 gives an arrow g.f : a1 -> a3. Composition is associative, and there's an "identity arrow" for every vertex, such that 1.f = f = f.1 for all arrows f. And that's it. A functor is a graph map that preserves composition and identities. A natural transformation is like a homotopy between functors: formally, if F and G are functors C -> D, a natural transformation \alpha : F -> G is a functor \alpha : 2 x C -> D, where 2 is the category (0 -> 1), and \alpha(0,-) = F, \alpha(1,-) = G. That isn't the usual definition, but it's equivalent to it.
- Every partially ordered set is a category: put a single arrow a -> b iff a ≤ b.
- Every group (in fact, every monoid) gives rise to a category: take one object, and an arrow for every element of the group, and compose as in the group.
- For every directed graph G, there's a category called the "free graph on G", which has an object for every vertex of G, and an arrow a -> b for every path in G from a to b. The identity arrows are the zero-length paths.
- There's a category Set, whose objects are sets and whose arrows are functions (we don't worry too much about set-theoretical size considerations, though it's often a good idea to assume that the arrows a -> b form a set for any objects a and b).
- Similarly, there are categories Grp, Mon, Rng, Top, Mfld, etc, of groups, monoids, rings, topological spaces, manifolds etc and the appropriate kind of map
- Recursively, there's a category Cat of categories (ignoring set theoretic hackery to avoid Russell's Paradox)
- Take any two categories C and D. There's a "functor category" [C,D} whose objects are functors C -> D and whose morphisms are natural transformations.
- Take the disjoint union of the categories formed from each symmetric group. The resulting category is equivalent (read "homotopic to") the category of finite sets and bijections.
- And so on...
Examples of functors:
- Take Grp, Top, etc, and forget the structure on the objects to give their underlying sets. Forget about any special properties of the morphisms and take the underlying function. This gives the "forgetful functor" to Set.
- To each directed graph, assign its free category. To each graph map f: G -> H, assign the obvious functor that it generates between the free categories on G and H. This gives the "free category" functor Digraph -> Cat. This is in a sense dual to the forgetful functor Cat -> Digraph.
- Similarly, there's a "free group" functor Set -> Grp, a "free ring" functor Set -> Rng, and so on. They're all dual to the appropriate forgetful functors.
I hope that helped :-)
... and, finally, the monkey performs in the free ring circus.