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Let R be a commutative ring. From R we can construct the category R\text{-Mod} of R-modules, which becomes a symmetric monoidal category when equipped with the tensor product of R-modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group \text{Pic}(R) of R is the group of isomorphism classes of R-modules which are invertible with respect to the tensor product.

By invertible we mean the following: for L \in \text{Pic}(R) there exists some L^{-1} such that the tensor product L \otimes_R L^{-1} is isomorphic to the identity for the tensor product, namely R.

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme \text{Spec } R).

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Let \Sigma_g be a closed orientable surface of genus g. (Below we will occasionally write \Sigma, omitting the genus.) Then its Euler characteristic \chi(\Sigma_g) = 2 - 2g is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be 2 - 2g, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

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In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in \mathbb{CP}^n. When n = 3 the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.

Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.

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Let C be a (locally small) category. Recall that any such category naturally admits a Yoneda embedding

\displaystyle Y : C \ni c \mapsto \text{Hom}(-, c) \in \widehat{C}

into its presheaf category \widehat{C} = [C^{op}, \text{Set}] (where we use [C, D] to denote the category of functors C \to D). The Yoneda lemma asserts in particular that Y is full and faithful, which justifies calling it an embedding.

When C is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.

Theorem: The Yoneda embedding Y : C \to \widehat{C} exhibits \widehat{C} as the free cocompletion of C in the sense that for any cocomplete category D, the restriction functor

\displaystyle Y^{\ast} : [\widehat{C}, D]_{\text{cocont}} \to [C, D]

from the category of cocontinuous functors \widehat{C} \to D to the category of functors C \to D is an equivalence. In particular, any functor C \to D extends (uniquely, up to natural isomorphism) to a cocontinuous functor \widehat{C} \to D, and all cocontinuous functors \widehat{C} \to D arise this way (up to natural isomorphism).

Colimits should be thought of as a general notion of gluing, so the above should be understood as the claim that \widehat{C} is the category obtained by “freely gluing together” the objects of C in a way dictated by the morphisms. This intuition is important when trying to understand the definition of, among other things, a simplicial set. A simplicial set is by definition a presheaf on a certain category, the simplex category, and the universal property above says that this means simplicial sets are obtained by “freely gluing together” simplices.

In this post we’ll content ourselves with meandering towards a proof of the above result. In a subsequent post we’ll give a sampling of applications.

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(This is an old post I never got around to finishing. It was originally going to have a second half about pointless topology; the interested reader can consult Vickers’ Topology via Logic on this theme.)

Standard presentations of propositional logic treat the Boolean operators “and,” “or,” and “not” as fundamental (e.g. these are the operators axiomatized by Boolean algebras). But from the point of view of category theory, arguably the most fundamental Boolean operator is “implies,” because it gives a collection of propositions the structure of a category, or more precisely a poset. We can endow the set of propositions with a morphism p \to q whenever p \Rightarrow q, and no morphisms otherwise. Then the identity morphisms \text{id}_p : p \to p simply reflect the fact that a proposition always implies itself, while composition of morphisms

\displaystyle (p \Rightarrow q) \wedge (q \Rightarrow r) \to (p \Rightarrow r)

is a familiar inference rule (hypothetical syllogism). Since it is possible to define “and,” “or,” and “not” in terms of “implies” in the Boolean setting, we might want to see what happens when we start from the perspective that propositional logic ought to be about certain posets and figure out how to recover the familiar operations from propositional logic by thinking about what their universal properties should be.

It turns out that when we do this, we don’t get ordinary propositional logic back in the sense that the posets we end up identifying are not just the Boolean algebras: instead we’ll get Heyting algebras, and the corresponding notion of logic we’ll get is intuitionistic logic.

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The goal of this post is to compute the cohomology of the n-torus X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is \mathbb{Z} by default. At the end we will interpret this computation in terms of cohomology operations.

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Previously we saw that Cantor’s theorem, the halting problem, and Russell’s paradox all employ the same diagonalization argument, which takes the following form. Let X be a set and let

\displaystyle f : X \times X \to 2

be a function. Then we can write down a function g : X \to 2 such that g(x) \neq f(x, x). If we curry f to obtain a function

\displaystyle \text{curry}(f) : X \to 2^X

it now follows that there cannot exist x \in X such that \text{curry}(f)(x) = g, since \text{curry}(f)(x)(x) = f(x, x) \neq g(x).

Currying is a fundamental notion. In mathematics, it is constantly implicitly used to talk about function spaces. In computer science, it is how some programming languages like Haskell describe functions which take multiple arguments: such a function is modeled as taking one argument and returning a function which takes further arguments. In type theory, it reproduces function types. In logic, it reproduces material implication.

Today we will discuss the appropriate categorical setting for understanding currying, namely that of cartesian closed categories. As an application of the formalism, we will prove the Lawvere fixed point theorem, which generalizes the argument behind Cantor’s theorem to cartesian closed categories.

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