Locally Presentable

Introduction Presentability is the right size condition for category theory: it is large enough to contain “all the categories that come up in nature” — modules over a ring spectrum, sheaves on a site, parametrised spectra, anima, etc. — yet small enough that the adjoint functor theorem applies and that $\mathrm{Hom}$-functors are controlled by an essentially small skeleton. The intuition is that a presentable category is generated by a small piece (its $\kappa$-compact objects) under filtered colimits. Everything else is recovered as a limit of these atoms. This is what makes the category “tame”: universal constructions exist, adjoint functors exist as soon as one side preserves the right kind of (co)limit, and almost all the categories of homotopy theory live here. ...