Homotopy colimits

A common technique when studying a category \$\mathcal{C}\$ is to study a localization \$\mathcal{C}[W^{-1}]\$ instead — this is what we’re doing when we study the homotopy category of spaces or the derived category of complexes in an abelian category. These localizations tend to have nice formal properties but can be slippery in other ways. For example, colimits in these settings are often not functorial! In this talk, we’ll see examples of this problem in topology, algebra, and geometry, and then discuss how the theory of homotopy colimits addresses this issue by giving us precise control over many of the localizations we care about. Finally, we’ll see how these homotopy-coherent constructions are best understood in the language of \$\infty\$-categories.