riehl2016-category-theory

Abstract

This book provides an introduction to the foundational concepts of category theory, covering categories, functors, natural transformations, limits, colimits, adjunctions, and Kan extensions. We emphasize the development of these concepts through concrete examples drawn from diverse areas of mathematics, including abstract algebra, algebraic topology, and geometry. By presenting categorical abstractions in their original mathematical contexts, we demonstrate how category theory unifies disparate fields and provides a rigorous language for formalizing structural analogies.

Introduction

Category theory is the study of mathematical analogy.

Categories, Functors, Natural Tranformations

Definition

A category $\mathcal{C}$ is made up of:

• A type of objects $|\mathcal{C}|:\mathrm{Set}$
• A type of hom-sets ($\mathcal{C}[\_,\_]:|\mathcal{C}|\to |\mathcal{C}|\to \mathrm{Set}$)
• $\mathrm{id}:\forall a\to \mathcal{C}[a,a]$ identity morphism
• $\circ :\mathcal{C}[b,c]\to \mathcal{C}[a,b]\to \mathcal{C}[a,c]$ composition operator Such that,
• ${\mathrm{id}}_a$ is a left- and right-inverse of $\circ$.
• $\circ$ is associative.

Link to original

Group Extension

Definition

Let $G,K:\mathrm{Ab}$ be an Abelian group. A group extension of $G$ by $K$ is a group $E$ that fits into a short exact sequence of the form:

$1\to K\stackrel{i}{\to }E\stackrel{p}{\to }G\to 1$

where $i$ is an injective homomorphism, $p$ is a surjective homomorphism, and $\text{im}(i)=\text{ker}(p)$. In this context, $K$ is isomorphic to a normal subgroup of $E$, and the quotient $E/K$ is isomorphic to $G$.

Equivalence of Extensions

Two extensions $E$ and $E^{\prime }$ of $G$ by $K$ are equivalent if there exists a group isomorphism $\varphi :E\to E^{\prime }$ such that the following diagram commutes:

1 \to K \to &E \to G \to 1 \\ \downarrow \text{id}_K \quad &\downarrow \phi \quad \downarrow \text{id}_G \\ 1 \to K \to &E' \to G \to 1 \end{aligned}$$ ### Classification Extensions are classified by the second [[cohomology group]] $H^2(G, K)$ when $K$ is Abelian. If the sequence [[Split Morphism (Category Theory)|splits]], meaning there exists a [[Section (Category Theory)|section]] $s: G \to E$ such that $p \circ s = \text{id}_G$, the extension is a *[[semidirect product]]* $K \rtimes_\theta G$. If the extension is central, $K$ lies in the center of $E$, denoted $Z(E)$. ## References 1. Brown, K. S. (1982). *Cohomology of Groups*. Springer-Verlag. 2. Rotman, J. J. (1995). *An Introduction to the Theory of Groups*. Graduate Texts in Mathematics.Link to original

Universal Properties, Representability and Yoneda

representation

Limits and Colimits

Adjunctions

Monads and their Algebras

All Conceptsx are Kan Extensions

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