A sheaf on a topological space $ X $ can be defined as a presheaf $ F: \text{Open}(X)^{\text{op}} \to \mathbf{Set} $ such that for any open covering $ { U_i } $ $ (i \in I) $ of an open set $ U $, the …

abstract. This project covers free semimodules and their examples. We define various algebraic structures via $\Omega$-algebras (from universal algebra) with an emphasis on semimodules. We continue to …

The classes of monomorphisms, epimorphisms, bimorphisms, split monomorphisms, and split epimorphisms are closed under composition and contain all isomorsphisms. To show these classes are closed under …

Let $\mathbb{C}$ be a fixed category. Consider a morphism $f:A\to B$ in $\mathbb{C}$ then, for any object $X$ in $\mathbb{C}$, …

The Yoneda lemma says that for a category $\mathbb{C}$, an object $C$ in $\mathbb{C}$, and a functor $S:\mathbb{C}\to\textbf{Sets}$, $\operatorname{Nat}(\operatorname{hom}(C,-),S)$ is in bijection …

Consider a vector space $V$, when $X$ is a basis of $V$, we say that $V$ is a free vector space on $X$ 1. For any vector space $V$ and $W$ and a basis $X$ of $V$, if there is a map $f:X\to W$ then …

An algebraic category is defined to be any full subcategory of $\textbf{Alg}(\Omega)$, the catgeory of all $\Omega$-algebras. We can define various algebraic structures as algebraic categories. For …

A monoid $(M,e,m)$, where $M$ is a set, $e$ an element of $M$, and $m$ an associative binary operation on $M$, can be viewed as a single object category. Take $M_0 = \{M\}$ or $M_0 = \emptyset$ and …

(Last updated: December 2022) I will be making a series of posts that contains edited versions of some of the short answers I submitted for my category theory course at the University of Cape Town. …