Definition. Let $[n]$ denote the set $\{0,1,\dots,n\}$. We define the category $\Delta\subseteq \mathcal{Set}$ (called the category of combinatorial simplices) as follows: The objects of $\Delta$ are the sets $[n]$. The morphisms in $\Delta$ are the order-preserving set functions. Remark. For each $n$, the set $[n]$ is a category using the linear ordering on $n$. To be precise,…
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 $p$-adic completion of $\bar{\mathbb{Q}}_p$ (denoted by $\mathbb{C}_p$). For those who are well-versed in the basics of $\mathbb{Q}_p$, I would recommend skipping to…
Definition. A (small) directed graph $G$ is a 3-tuple $(V,E,d)$ where $V$ and $E$ are sets and $d:E\to V\times V$. Similarly, a subgraph $A$ of $G$ is a 3-tuple $(V_A,E_A,d)$ such that $E_A$ is contained in $E$ and $V_A\times V_A\subseteq V\times V$ contains $d(E_A)$. Definition. If $V\times V$ is replaced with $V\times V / \{(x,y)\sim (y,x)\}$,…