Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F (Definition 1.11) from G to its set of connected components (seen as a discrete category). From the observation that E and F differ unless G[x, x] = {idx} for every object x of G, we prove there is a fibered equivalence (Definition 1.12) from C[Σ-1] (Proposition 1.1) to C/Σ (Proposit...

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined ‘in parallel’. This chapter introduces the theory of monoidal categories, and shows how our example categories Hilb, Set and Rel can be given a monoidal structure. We also introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work ...

For a locally presentable abelian category B with projective generator, we construct the derived and contraderived model structures on of complexes, proving in particular existence enough homotopy complexes objects. We also show that D(B) is generated, as triangulated coproducts, by generator B. Grothendieck A, injective coderived complexes. Assuming Vopěnka's principle, prove D(A) products, co...

Most natural domains can be represented in multiple ways: we can categorize foods in terms of their nutritional content or social role, animals in terms of their taxonomic groupings or their ecological niches, and musical instruments in terms of their taxonomic categories or social uses. Previous approaches to modeling human categorization have largely ignored the problem of cross-categorizatio...