In this paper we obtain the Cartier duality for k-schemes of commutative monoids functorially without providing the vector spaces of functions with a topology (as in [DGr, Exposé VIIB by P. Gabriel, 2.2.1]), generalizing a result for finite commutative algebraic groups by M. Demazure & P. Gabriel ([DG, II, §1, 2.10]). All functors we consider are functors defined over the category of commutativ...

By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed m...

The goal of this paper is construct a framework which can be used to proof results of this kind for a wide class of closed model categories and functors between those categories. Originally I was interested in the proof that the symmetric power functors respect A-equivalences between simplicial schemes but it soon became clear that similar problems arise for other categories (such as finite cor...

Higher-kinded polymorphism —i.e. abstraction over type constructors— is an essential component of many functional programming techniques such as monads, folds, and embedded DSLs. ML-family languages typically support a form of abstraction over type constructors using functors, but the separation between the core language and the module language leads to awkwardness as functors proliferate. We s...

The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of H...

ion. It is more a descent to the mesosphere. For simplicity of discussion, we shall stick to biset functors, though variants of the material apply also to other kinds of group functors. In our account of biset functors, we mentioned that the abstraction here is rather similar to the abstraction behind the notion of ratios: just as the Greeks moved towards abstract ratios in place of ratios of p...

We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of ex...

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A ∞ -functor Y : A → A ∞ (A,C) is a full embedding for an arbitrary unital A ∞ -category A. For a differential graded k-quiver B we define the free A ∞ -category FB generated by B. The main result is that the restriction A ∞ -functor A ∞ (FB,A) → A1(B,A) is an equivalence, where objects of the l...

We present a simple way of incorporating the structure contextual extensions into quantum gravity models. The [Formula: see text]-algebras, originally proposed for hidden variables, are generalized to cones indexed by contexts and their limit in category. By abstracting models as functors, we study categorical limits these functors several Such useful building topos-theoretic gravity.

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne’s determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universa...