We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In or...

We show that a derivator is stable if and only homotopy finite limits colimits commute, limit functors have right adjoints, colimit left adjoints. These characterizations generalize to an abstract notion of "stability relative class functors", which includes in particular pointedness, semiadditivity, ordinary stability. To prove them, we develop the theory derivators enriched over monoidal weig...

The Basic Mereology framework of [8] is enriched by adding colimit construction from Category Theory. The new framework is then used to model component-based software architecture.

We establish sufficient and necessary conditions for a natural sink to be a colimit in DCPO. Based on these conditions we show how to construct a colimit for any functor F from a small category „ into the category DCPO. This demonstrates that the category DCPO is cocomplete. We also investigate under what conditions the colimit object is algebraic. 0 Introduction Colimits play an important role...

This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate ‘nerves of C’ are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major resul...

We extend to the non-connective case a lemma of Bökstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH.

We introduce a general construction on 2-monads. develop background maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we the colimit induced by map show that obtain structure 2-monad give characterisation its algebras. Finally, apply to 2-monads between free symmetric monoidal cartesian combine them into linear-non-linear 2-monad.