So 5am rolls around, and I’m still not asleep. Of course. It’s not like I have to be up in 2.5 hours or anything. My brain is conspiring against me. Whilst rolling around in bed NOT SLEEPING, my thoughts turned to sheaves: just what are they? I posted a bit ago (longer than I like)… Continue reading Sheaves on my mind
As the title says, I want to show that for any topological space , the category of set-valued presheaves PSh(X) on has all finite limits and co-limits. First, PSh(X) has both initial and terminal objects. With a bit of thought, these are (obviously) the constant functors 0 and 1 (resp.) where, for all open subsets of we have and… Continue reading Presheaves of Sets are (finitely) Bi-Complete
For anyone who has done a bit of work in modern geometry (primarily the notion of a (smooth) manifold), we want objects to be “locally” trivial, or easy to study. The global structure might be this crazy awesome geometric shape, but locally it’s going to look like boring old or something like that. How much… Continue reading What are Sheaves, and why should I care?
Sorry for the delay since my last post (to those who actually read this…) So I stumbled across a really nice way of looking at universal properties that is equivalent to specifying them as a terminal (or initial) object in a suitable comma category, but it has a much nicer “intuitive feel.” Cones (and co-Cones)… Continue reading Universal Properties IV: Cones and a first look at Limits
So last time I mentioned that we could describe the kernel of a group homomorphism via a universal property. For example, let be a group homomorphism, and let D be the full subcategory of Grp consisting of all groups such that for any group homomorphism we have is the zero homomorphism from to . Good. Now if A… Continue reading Universal Properties III: Bringing it all together
In my last post, I spent a good bit trying to get you interested in looking at universal properties. Hopefully, you’ve read that post, and are still sufficiently interested to continue, because it’s only going to get harder before we see the light. We left off at defining these special objects in some category C called… Continue reading Universal Properties II: Comma Categories
So I want to take some time to talk about universal properties. I personally think they’re awesome because if you look hard enough, you start to see them everywhere in mathematics. Especially in abstract algebra and algebraic geometry. They admit a fairly intuitive explanation, but the actual details of their definition require a lot of work.… Continue reading Universal Properties: a Prelude
Hello again! Last time I got to talking about these mathematical things called “categories.” If you’ve ever taken a class in higher math, whatever that means, you should know by now that whenever we define a new mathematical object, the next step is to define what it means to talk about “functions” between them. In… Continue reading Functors and Natural Transformations!
Hello! I’m a terrible writer, so I’m going to dive right it. My selfish goal is to gain a thorough understanding of category theory, but that road is not a straight shot. It requires a great deal of knowledge and experience from all of mathematics to really grok many of the abstract methods employed. This… Continue reading Hello & Here we go