Filtered colimits, i.e., colimits over schemes D such that D-colimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that D-colimits in Set commute with finite products. An important example: reflexive coequalizers are sifted colimits. Generalized varieties are defined as free completions of small categories under sift...

We introduce flagged (∞, n)-categories and prove that they are equivalent to Segal sheaves on Joyal’s category Θn. As such, flagged (∞, n)-categories provide a model-independent formulation of Segal sheaves. This result generalizes the statement that n-groupoid objects in spaces are effective, as we explain and contextualize. Along the way, we establish a useful expression for the univalent-com...

Higher Homotopy van Kampen Theorems allow some colimit calculations of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one res...

Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451-479) are not correct. We show that their use can be avoided and all remaining results remain correct. Propositions 4.2 and 4.3 of the author’s [3] are not correct and I am grateful to J. F. Jardine for pointing it out. In fact, consider the diagram D sending the one morphism category to the point ∆0 in the homo...

In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH)...

Let $S$ be a monoid. In this paper, we prove every class of $S$-acts having a flatness property is closed underdirected colimits, it extends some known results. Furthermore thisresult implies that every $S$-act has a flatness cover if and only if it has a flatness precover.

Morava E-theory is a much-studied theory important in understanding K(n)-local stable homotopy theory, but it is not a homology theory in the usual sense. The object of this paper is to show that the usual computational methods, spectral sequences, used to compute with homology theories also apply to Morava E-theory. Conceptually, we show that Morava E-theory is n derived functors away from bei...

This paper aims to bring the benefits of the use of Category Theory to the field of Semantic Web, where the coexistence of intrinsically different models of local knowledge makes difficult the exchanging of information. The paper uses categorical limit and colimit to define operations of breaking and composing ontologies, formalizing usual concepts in ontologies (alignment, merge, integration, ...

Parametrized iterativity of an algebra means the existence of unique solutions of all finitary recursive systems of equations where recursion is allowed to use only some variables (chosen as a parameter). We show how such algebras can be introduced in an arbitrary category A by employing a base, i.e., an operation interpreting objects of A as monads on A. For every base we prove that free base ...

In this paper we introduce a new setting, based on partial algebras, for studying constructions of finitely generated free algebras. We give sufficient conditions under which the finitely generated free algebras for a variety V may be described as the colimit of a chain of finite partial algebras obtained by repeated application of a functor. In particular, our method encompasses the constructi...