Public Notes

Some notes I write.

A Compendium on Basic Commutative Ring Structures

Abstract: It seems that a majority of my undergraduate algebra career was spent trying to prove inclusions, mutual intersections, and reformulations of different commutative ring structures. In this short story, I will attempt to provide a compendium of equivalency theorems, properties, and other elements of commutative algebra. The pedagogy here is purposefully unmotivating; you can probably…

Visualizing Simplicial Objects

Definition. Let [latex][n][/latex] denote the set [latex]\{0,1,\dots,n\}[/latex]. We define the category [latex]\Delta\subseteq \mathcal{Set}[/latex] (called the category of combinatorial simplices) as follows: The objects of [latex]\Delta[/latex] are the sets [latex][n][/latex]. The morphisms in [latex]\Delta[/latex] are the order-preserving set functions. Remark. For each [latex]n[/latex], the set [latex][n][/latex] is a category using the linear ordering on [latex]n[/latex]. To be precise,…

Failure of the Hahn-Banach in ℂₚ

The Hahn-Banach is known to be a crucial tool in functional analysis. The following are lecture notes written a while ago for my functional analysis course on the failure of Hahn-Banach in the [latex]p[/latex]-adic completion of [latex]\bar{\mathbb{Q}}_p[/latex] (denoted by [latex]\mathbb{C}_p[/latex]). For those who are well-versed in the basics of [latex]\mathbb{Q}_p[/latex], I would recommend skipping to…

Diagrams, Categorical Limits, and Topoi (Part 1)

Definition. A (small) directed graph [latex]G[/latex] is a 3-tuple [latex](V,E,d)[/latex] where [latex]V[/latex] and [latex]E[/latex] are sets and [latex]d:E\to V\times V[/latex]. Similarly, a subgraph [latex]A[/latex] of [latex]G[/latex] is a 3-tuple [latex](V_A,E_A,d)[/latex] such that [latex]E_A[/latex] is contained in [latex]E[/latex] and [latex]V_A\times V_A\subseteq V\times V[/latex] contains [latex]d(E_A)[/latex]. Definition. If [latex]V\times V[/latex] is replaced with [latex]V\times V / \{(x,y)\sim (y,x)\}[/latex],…