Gabriel showed that for a unital ring R, there exists bijective correspondence between the set of filters R and Giraud subcategories Mod(R). In this paper we prove result analogous to one given by Gabriel: small preadditive category C, left C Mod(C).

We prove that Morel-Voevodsky's stable $\mathbb{A}^1$-homotopy theory affords the universal coefficient system, giving rise to Grothendieck's six operations.

Essential to ML-style module systems is the concept of a functor, i.e. a function from modules to modules. Functors are ML’s way of providing generic data structures and reusable program components. Yet, there is a bewildering schism in ML land regarding the actual semantics of functors, familiar to experts under the code words generative vs. applicative. In Standard ML [3], each application of...

We provide a more economical refined version of Evrard’s categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which satisfies the (dual) assumption of Quillen’s Theorem B. Introduction The problem treated here is how to replace a functor f : C→ D between small cat...

We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this ...

The C*-algebra of bounded operators on the separable Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through `∞(X). Consequently, there is no faithful functor discretizing C*-algebras to W*-algebras this way.