Presheaves of Sets are (finitely) Bi-Complete

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 andContinue reading “Presheaves of Sets are (finitely) Bi-Complete”

What are Sheaves, and why should I care?

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 muchContinue reading “What are Sheaves, and why should I care?”

Universal Properties IV: Cones and a first look at Limits

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”

Universal Properties III: Bringing it all together

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 AContinue reading “Universal Properties III: Bringing it all together”

Universal Properties II: Comma Categories

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 calledContinue reading “Universal Properties II: Comma Categories”

Universal Properties: a Prelude

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”

Functors and Natural Transformations!

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.  InContinue reading “Functors and Natural Transformations!”

Welcome, stay, & learn category theory with me

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.  ThisContinue reading “Welcome, stay, & learn category theory with me”