Abstract Nonsense

Crushing one theorem at a time

Smooth Maps and the Category of Smooth Manifolds (Pt. III)


Point of Post: This is a continuation of this post.

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September 3, 2012 Posted by | Manifold Theory, Topology | , , , , , , | Leave a comment

Smooth Maps and the Category of Smooth Manifolds (Pt. II)


Point of Post: This is a continuation of this post.

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September 3, 2012 Posted by | Manifold Theory, Topology | , , , , , | 1 Comment

Smooth Maps and the Category of Smooth Manifolds (Pt. I)


Point of Post: In this post we define what it means for a map between two manifolds to be smooth.

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Motivation

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We literally defined smooth manifolds to be the topological spaces where we will have a relatively sound meaning of what a “smooth map” is. Thus, it would seem that the first order of business is to fully define and explore this notion of smooth map. The basic idea though is precisely what we have said before. A map between smooth manifold will be smooth if it is smooth locally around each point and its image–when we think about the space locally as Euclidean space.

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The interesting part is that once we define smooth map we will then be able to define the category of (finite dimensional) smooth manifolds. We will then be able to discuss the functor which takes a smooth manifold to it’s algebra of smooth functions (for us, function will mean a map into \mathbb{R}). We will then be able to make sense of the following statement: the smooth structure of a manifold  is largely encoded in its algebra of smooth functions.

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September 3, 2012 Posted by | Manifold Theory, Topology, Uncategorized | , , , , , | 2 Comments

The Fundamental Groupoid and Group (Pt. IV)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 4 Comments

The Fundamental Groupoid and Group (Pt. III)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 1 Comment

The Fundamental Groupoid and Group (Pt. II)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 1 Comment

The Fundamental Groupoid and Group (Pt. I)


Point of Post: In this post we describe the fundamental group and groupoid functors.

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Motivation

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We have discussed in previous posts how the notion of homotopy as being an equivalence that captures, and focuses in on, the various notions of connectivity that a space carries. Now, the first thing one does when one defines a new notion of equivalence is to try to classify all objects of interest up to this equivalence. For us, this means that we would (in a perfect world) be able to classify all spaces up to homotopy equivalence. Of course, as in most categories(subjects of study), this is an untenable goal. It is not practically possible to, say, explicitly classify all finite groups (even though we have made stupendous strides in this direction).

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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 2 Comments

Left Exact, Right Exact, and Exact Functors


Point of Post: In this post we discuss the notion of left exact, right exact, and exact functors between abelian categories–giving several equivalent definitions.

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Motivation

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We finally get back to the homological algebra side of things. What we would now like to discuss is just a thinly veiled continuation of our last post on continuous and cocontinuous functors. Namely, we are going to ask when a functor preserves/partially preserves the exactness of a given short exact sequence in some abelian category. This is of absolutely fundamental importance in homological algebra because as we have previously stated the degree to which a sequence fails to be exact is the reason we study homological algebra and thus it behooves us to figure out when applying a functor to an exact sequence introduces no new obstruction. In fact, a huge part of the homological algebra to come will be measuring how badly a certain type of functor fails to be exact.

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April 24, 2012 Posted by | Algebra, Category Theory, Homological Algebra | , , , | 3 Comments

Continuous and Cocontinuous Functors (Pt. II)


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April 15, 2012 Posted by | Algebra, Category Theory | , , , , , , , | 1 Comment

Continuous and Cocontinuous Functors


Point of Post: In this post we discuss the notion of continuous functors as

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Motivation

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Anyone doing math for very long, especially more abstract algebra or topology, is well-acquainted with the fact that constructions “commuting” often makes life very easy. For example, it is a hell of a lot of work to show from first principles that \pi_1(\mathbb{S}^1)\cong\mathbb{Z}. After reading such a proof the idea of finding \pi_1(\mathbb{S}^n) for n>1 may sound groan-worthy. That is until someone tells you the sigh-inducing fact that \pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y). Similarly, trying to compute \mathbb{Q}^n\otimes_\mathbb{Z}(\mathbb{Z}^r\oplus\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_m}) sounds gross until someone points out that M\otimes(N\oplus L)\cong (M\otimes N)\oplus( M\otimes L) and similarly for the other coordinate. In general the ability to commute certain constructions is what enables us to make certain fundamental computations and use these to compute things which are constructed from these fundamental parts.

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Of course, when we say “constructions” we secretly mean some kind of limit or colimit and the things they are commuting with is some kind of functor. Thus, it seems to behoove us, purely for the computational-theorem-proving-goodness involved, to talk about functors which commute with limits (called continuous functors) and those that commute with colimits (called cocontinuous functors).

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April 15, 2012 Posted by | Algebra, Category Theory | , , , , , , , , | 2 Comments