Unity of Opposites

A monad is a monoid in a category of endofunctors. In the previous post, we have established a very general notion of monoid which is suitable for many applications. For example, the category of chain complexes $R$-modules can be equipped with a monoidal product which associates to each pair $(C,C')$ of chain complexes the total complex $Tot(C\otimes C')$ of the double complex $(C\otimes C')_{i,j}= C_i\otimes C_j'$. The unit of this monoidal product is the complex with $R$ in dimension $0$ and $0$'s in all other dimension. A DG-algebra is precisely a monoid with respect to this monoidal product. For a given category $\mathcal{C}$, the functor category $[\mathcal{C};\mathcal{C}]$ is naturally equipped with thestructure of a strict monoidal category, whose monoidal product $\otimes$ is precisely functor composition, $F\otimes G= F\circ G$; and whose unit $1$ is precisely the identity functor $id_{\mathcal{C}}$. It is easily verified that this is functorial simulta...

Let $X$ be a topological manifold. Theorem: Let $x$ be a point in $X$. There is a small open neighborhood $U$ of $x$ such that for every $x'$ in $U$, there is a homeomorphism of $X$ with itself carrying $x'$ to $x$. Actually, we can say something substantially stronger, but it's a bit more subtle and harder to digest. Theorem: Let $x$ be a point in $X$. There is a small open neighborhood $U$ of $x$, such that for each $x'\in U$, there exists a homeomorphism $f_{x'}$ from $X$ to itself, carrying $x'$ to $x$; moreover, this family of homeomorphisms can be chosen such that the disjoint union $f= \coprod_{x'\in U}f_{x'} : U\times X\to U\times X$ is a homeomorphism of $U\times X$ with itself, carrying $(x', y)$ to $(x',f_{x'}(y))$, and in particular sending $(x',x')$ to $(x',x)$. The proof is an elegant geometrical construction. Let $\delta(U)\subset U\times X$ be the set of ordered pairs $(x,x)$ with $x\in U$. Then the homeomor...

An undergrad that I was a T.A. for recently asked me what my research is in. I decided to write up an answer as a blog post, in a way that is accessible to the average early undergrad student in STEM. I study logic. More specifically, I study semantics. In linear algebra, you start off with a certain formal algebraic language, the language of vector spaces - it has certain distinguished constant symbols you are permitted to use, like "0" for the zero vector, or any real number; you are permitted to use variables v, w, ... that range across the vectors; there are also special symbols that represent functions, like "+", for addition. There are also certain axioms, written in the language. One also has a list of formal logical laws that allow you to combine the axioms in various ways to derive theorems in the theory of linear algebra. Linear algebra from this point of view is the collection of all formal statements in the language of linear algebra that c...

In the last blog post I promised to introduce monads. Now that I have whetted your appetite, I will pull a bait and switch. We will not talk about monads; rather I will set the stage with some abstract nonsense. There is a famous definition of monad as a "monoid in the category of endofunctors $[\mathcal{C},\mathcal{C}]$ of some category $\mathcal{C}$." Before it is possible to parse this definition, we must understand what a monoid is. Along the way we will start to pick up the significance and substance of this definition, and why it should matter. There is a close connection between monoids and cosimplicial objects, which is what this post is really about; so before we even talk about monads properly, we will develop an inkling of why they would be relevant to simplicial theory. Be warned before reading that this is very heady stuff. It is something I avoided learning for a long time, as the section in Categories for the Working Mathematician on this material I found to...

I will quote at length from Godement to begin with. We will start by developing a cross product in sheaf cohomology, similar to the one present in singular cohomology. Let \(X,Y\) be a pair of spaces. We will work with sheaves of Abelian groups over \(X,Y\) and \(X\times Y\); everything we are saying of course is easily adaptable to \(A\)-modules for a commutative ring \(A\). If \(\mathcal{F}\) is a sheaf of Abelian groups  over \(X\) and \(\mathcal{G}\) is a sheaf of Abelian groups (henceforth simply "sheaf") over \(Y\), then it is possible to construct over \(X\times Y\) a "total tensor product sheaf" \(\mathcal{F}\hat{\otimes}\mathcal{G}\). (We use the hat to distinguish it from a tensor product of sheaves over a single space.) The definition of this sheaf is by the standard process: we will define a bundle \(\pi : B\to X\times Y\), which will turn out to be espace étalé of the sheaf \(\mathcal{F}\hat{\otimes}\mathcal{G}\). We construct \(B\) by defining the fib...