Reading about Topoi is rough going. The references I’ve found so far are seriously rough going. So instead of diving right in, I’m going to take a couple of steps back, to some of the foundational material that I think helps make it easier to see where the category theory is coming from. (As a general statement, I find that category theory is fascinating, but it’s so abstract that you really need to do some work to ground it in a way that makes sense. Even then, it’s not easy to grasp, but it’s worth the effort!)<\/p>\n
A lot of category theoretic concepts originated in algebraic topology. Topoi follows that – one of its foundational concepts is related to the topological idea of a sheaf<\/em>. So we’re going to start by looking at what a sheaf is.<\/p>\n <\/p>\n Suppose we’ve got a topological space. A topological space is, intuitively, a collection of objects called points<\/em>, and a relationship that defines “nearness”. If you pick any point in the space, then you can define an ever-expanding circle of neighborhoods<\/em>, where being in a neighborhood of a point means, roughly, that you’re no further<\/em> from that point than any of the other points in the neighborhood. There’s a lot more to it than that, but that’s a good intuitive start.<\/p>\n In basic topology, we mostly focus either vary narrowly on local properties of a topological space (for example, when we study manifolds, which are objects that on a small, local scale look like they’re euclidean, even though on a large scale, they may not be), or they focus on global properties of the entire space. Sheafs are a mechanism that can allow us to bridge those two disparate universes, and find the connection between the local view and the global view.<\/p>\n A sheaf is a very general kind of structure that provides ways of mapping or relating local information about a topological space to global information about that space. There are many different kinds of sheaves; rather than being exhaustive, I’ll pretty much stick to a simple sheaf of functions on the topological space. Sheaves show up all over<\/em> the place, in everything from abstract algebra to algebraic geometry to number theory to analysis to differential calculus – pretty much every major abstract area of mathematics uses sheaves.<\/p>\n Since what I’m ultimately interested in Topoi, I’m going to stick to a simple categorical definition of a sheaf of continuous functions on a topological space. To figure out what a sheaf is, we’ll start with something weaker: a presheaf<\/em>. <\/p>\n A presheaf<\/em> That’s one hell of a mouthful, so let’s pick it apart a bit.<\/p>\n The restriction morphisms are the important thing in the definition. What we’re doing with the presheaf is providing a way of taking some statement about The properties of the restriction morphisms basically say:<\/p>\n So, basically, we’ve got a mapping from open sets in a space to sets, and that mapping is guaranteed to preserve the structure of the topological space. Why should we care about that? Because we can map it to sets that describe things about the space in ways that are easier to understand that the space itself. Topological spaces can be nightmarish beasts to work with; we can instead look at simpler sets. But because the presheaf gives us a mapping to the simpler sets that’s guaranteed to keep the structure of the topological space, we can safely work with the simpler objects.<\/p>\n There’s also another way of describing a presheaf in terms of category theory which I hadn’t seen before, but which I discovered via Wolfram’s Mathworld<\/a> while I was researching this post. It’s based on the idea of contravariant functors<\/em>.<\/p>\n This is going to get a bit dense, but bear with me. It does, eventually, make sense.<\/p>\n
\n
\n
\n
and 
. A covariant functor 
to an object 
.<\/li>\n
to a morphism 
.<\/li>\n<\/ol>\n
– that is, every identity arrow of an object in 
– if you compose 
and 
in 
.<\/p>\n
from 
, every function 
will be mapped to a function whose results<\/em> are acceptable as inputs to the mappings of things that used to compose with f.<\/p>\n
, where the objects in the category are the open sets of 
, there is an arrow 
if\/f 
.<\/p>\n