Category Theory

Introduction Compactly assembled categories are the right generality for modern applications where compact generation fails but the category still feels “finite-dimensional”. The motivating example: $\mathsf{Shv}(X)$ for $X$ locally compact Hausdorff is not compactly generated, yet behaves like a compactly generated category for nearly every K-theoretic and trace-theoretic purpose. The intuition: an object of a compactly assembled category is built out of compactly exhaustible atoms — sequences whose transition maps are “compact” in a precise sense. Compactness is now a property of morphisms, not just objects. In the stable world, this generality is forced on us: compactly assembled is precisely dualizable in $\mathsf{Pr}^L_{\mathrm{st}}$, which is what we need for K-theory, traces, and the symmetric monoidal structure to behave. ...

Introduction Compactly generated categories are the tame end of the presentable spectrum. Every object is a filtered colimit of compact ones, and most homological invariants — K-theory, Hochschild homology, cyclic homology — are determined by the small subcategory of compact objects. The idea: a “big” category is recovered, up to filtered colimits, from a small skeleton of finite/perfect/dualizable atoms. Examples are everywhere in algebra and topology: spectra (compact = finite spectra), modules over a ring spectrum (compact = perfect), the derived category of a ring (compact = perfect complexes). The fact that all of homotopy theory’s “large” categories of interest are compactly generated is why the theory of K-theory, traces and dimensions is as workable as it is. ...

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. ...