Adámek, Herrlich, and Reiterman showed that a cocomplete category A is cocomplete if there exists a small (full) subcategory B such that every A-object is a colimit of B-objects. The authors of the present paper strengthened the result to totality in the sense of Street and Walters. Here we weaken the hypothesis, assuming only that the colimit closure is attained by transfinite iteration of the...

We establish two versions of a central theorem, the Family Colimit Theorem, for coarse coherence property metric spaces. This is geometric and so well-defined finitely generated groups with word metrics. It known that fundamental group has important implications classification high-dimensional manifolds. The Theorem one permanence theorems give structure to class coarsely coherent groups. In fa...

The only proof of this I know is kind of fiddly and not enlightening. But the point is that in most categories that appear ‘in nature,’ regular, strong, and extremal epics are about the same, and all are the correct notion of quotient, while ordinary epimorphisms may not be. For example, N→ Q is epic in Rings, but not extremal epic. In Top the epics are the surjective maps, while the regular = ...

We give substance to the motto “every partial algebra is the colimit of its total subalgebras” by proving it for partial Boolean algebras (including orthomodular lattices), the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. Both pairs of results are related by taking projections. As cor...

We present a general approach for representing and combining alignments and computing these combinations, based on the category theoretic notions of diagram, pushout, and colimit. This generalises the possible ‘shapes’ of alignments that have been introduced previously in similar approaches. We use the theory of institutions to represent heterogeneous ontologies, and show how the tool Hets can ...

Limits and colimits of diagrams, defined by maps between sets, are universal constructions fundamental in different mathematical domains and key concepts in theoretical computer science. Its importance in semantic modeling is described by M. Makkai and R. Paré in [1], where it is formally shown that every axiomatizable theory in classical infinitary logic can be specified using diagrams defined...