• homological algebra
• higher algebra
• higher category theory

Derived algebraic geometry has seen its beginnings in the ‘90s as a generalization of algebraic geometry in which one seeks to replace triangulated categories of interest, like the triangulated derived category of quasi-coherent sheaves on a scheme, with higher categorical enhancements. In many examples, existence of enhancements is clear, but the uniqueness of these is less so.

In this thesis we examine the problem of establishing when uniqueness of the enhancement holds.

The first chapter covers the basics of simplicial sets and $\infty$-categories, highlighting the similarities with the theory of ordinary categories rather than the combinatorical intricacies.

In the second chapter we introduce triangulated categories and construct the main examples of these, i.e. derived categories of abelian categories.

In the third chapter we follow Lurie in \cite{LurieHA17} and define stable and prestable $\infty$-categories, then we proceed to construct dg-enhancements of some derived categories, and translate them to stable enhancements through the dg-nerve construction. In the last part, following B.Antieau in [1], we prove uniqueness of the stable enhancement of $T$ under the following technical conditions: $\mathcal{A}$ is an abelian category with enough injectives and $T=\mathrm{D}^+(\mathcal{A})$ (Lemma 3.10.2); $\mathcal{A}$ is a Grothendieck abelian category and $T=\mathrm{D}(\mathcal{A})$ such that the enhancement is presentable (Theorem A), or $T$ is compactly generated (Theorem B). Finally, we sketch a method developed by Canonaco, Neeman and Stellari in [5] that clarifies and generalizes these results.