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We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest, such as finite sets, Boolean functions, topological spaces, vector spaces, semi-linear and semi-algebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of non-uniform computational complexity (such as circuit complexity), while on the other hand it has features which make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes., Comment: 47 pages

We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting.

We provide descriptions of the derived categories of degree $d$ hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem of Orlov. Using a local generator and Morita theory, we re-interpret the resulting matrix factorization category as a derived-equivalent sheaf of dg-algebras on the base. Then, applying homological perturbation methods, we obtain a sheaf of $A_\infty$-algebras which gives a new description of homological projective duals for (relative) $d$-Veronese embeddings, recovering the sheaf of Clifford algebras obtained by Kuznetsov in the case when $d=2$., Comment: 30 pages, expanded from arXiv:1306.3957, submitted

We provide a geometric approach to constructing Lefschetz collections and Landau-Ginzburg Homological Projective Duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese embeddings in the setting of Landau Ginzburg models. Our results also extend to a relative Homological Projective Duality framework., Comment: 32 pages, expanded the original into two parts, accepted for publication in JEMS

Generalizing Eisenbud's matrix factorizations, we define factorization categories. Following work of Positselski, we define their associated derived categories. We construct specific resolutions of factorizations built from a choice of resolutions of their components. We use these resolutions to lift fully-faithfulness statements from derived categories of Abelian categories to derived categories of factorizations and to construct a spectral sequence computing the morphism spaces in the derived categories of factorizations from Ext-groups of their components in the underlying Abelian category., Comment: v2: Expanded further for enjoyment. 41 pages. v1: 24 pages, expanded from a section in arXiv:1203.6643, comments welcome

We prove an equivalence between the derived category of a variety and the equivariant/graded singularity category of a corresponding singular variety. The equivalence also holds at the dg level., Comment: 16 pages