We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.

Characterising colimiting ω-cocones of projection pairs in terms of least upper bounds of their embeddings and projections is important to the solution of recursive domain equations. We present a universal characterisation of this local property as ω-cocontinuity of locally continuous functors. We present a straightforward proof using the enriched Yoneda embedding. The proof can be generalised ...

We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibraction categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a right derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for left derivators. Th...

We investigate when Isomorphism Conjectures, such as the ones due to BaumConnes, 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 disp...

The goal of this paper is to introduce methods that allow us to calculate certain limits and colimits in the category of modules over a principal ideal domain. We start with a quick review of basic categorical language and duality. Then we develop the concept of universal morphisms and derive limits and colimits as special cases. The completeness and cocompleteness theorems give us methods to c...