We introduce, comment and develop the Scott adjunction, mostly from point of view a category theorist. Besides its technical conceptual aspects, in nutshell we provide categorification topology over posets with directed suprema. From establish an adjunction between accessible categories colimits Grothendieck topoi. show that bicategory topoi is enriched $2$-category it has tensors respect to th...

It is shown that for any commutative unital ring R the category HopfR of R–Hopf algebras is locally presentable and a coreflective subcategory of the category BialgR of R–bialgebras, admitting cofree Hopf algebras over arbitrary R–algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be ...

If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by D-coalgebra objects of C. Let Rep(C,D) denote the category of all such functors, a full subcategory of ...

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have dis...

Morava E-theory E∨ n∗(−) is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava E-theory of a coproduct (resp. filtered homotopy colimit). The E2 term of this spectral sequence involves the der...

This article is an overview of the recent developments of a theory originally introduced by Leifer and Milner: given a formalism with a reduction semantics, a canonical labelled transition system is derived on which bisimilarity as well as other other equivalences are congruences, provided that the contexts of the formalism form the arrows of a category which has certain colimits. We shall also...

As usual, we consider integral operators attached to compactly supported measures η on the group G, and exploit the intrinsic sense of such operators on any reasonable representation space for G, for example, Hilbert, Banach, Fréchet, and LF (strict colimits of Fréchet), or, generally,, quasi-complete, locally convex spaces. For a representation π, V of G, and a compactly-supported measure η, t...

Suppose that A is a pointed CW-complex. We look at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complication of B. Studying the complication leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X .