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?”

Adjoint Functors

So last post I gave a (hurried) description of why adjoint pairs of functors are linked to this notion of “similar structure” between two categories.  In this post, I want to relate adjunctions to universal properties, and ultimately why we like adjoint pairs so much.     Say we’re working with the “free group onContinue reading “Adjoint Functors”

Isomorphism, Equivalence, and Adjunction

Isomorphism As mathematicians are wont to do, whenever we have a collection of algebraic “objects,” we want to know how to “relate” them.  In the case of categories, we saw earlier that maps called functors are what we want to examine. The next step after defining these structure preserving maps is to wonder what it meansContinue reading “Isomorphism, Equivalence, and Adjunction”

Limits and co-Limits: Some Cool Things

I’m not going to reiterate the definitions of limits and co-limits from the last post, so just look them up if you’re new here.  They’re not too hard. This post is mainly just about some random cool things I’ve noticed/ “remembered” / come across whilst playing with the notions of limit and co-limit in variousContinue reading “Limits and co-Limits: Some Cool Things”

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”