Notes on Moduli theory, Stacks and 2-Yoneda's Lemma

TL;DR

This paper surveys moduli theory and explains how stacks, via 2-category theory and a 2-Yoneda's lemma, address non-representability issues in moduli functors, providing foundational insights.

Contribution

It introduces a 2-categorical perspective on moduli problems, highlighting how stacks resolve non-representability through a 2-Yoneda's lemma.

Findings

Stacks circumvent non-representability of moduli functors

Introduction of 2-Yoneda's lemma in the context of stacks

Provides foundational understanding of 2-category theory in moduli problems

Abstract

This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purposes of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor $\mathcal{F}$ by using the 2-category of stacks. To this end, we shall briefly revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma.