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 show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is...

An algebraically exact category is one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small lim...

Goguen emphasized long ago that colimits are how to compose systems [7]. This paper corroborates and elaborates Goguen’s vision by presenting a variety of situations in which colimits can be mechanically applied to support software development by refinement. We illustrate the use of colimits to support automated datatype refinement, algorithm design, aspect weaving, and security policy enforcem...

There is a 2-category J -Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from J -Colim to the 2-category Cat of small categories is known to be monadic. We extend this result by considering not jus...

We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is indexed homotopy limits and colimits, which specializes to $G$-colimits $G$ finite group when chosen be orbit $G$. apply this show that $G$-$\infty$-category $G$-spaces freely generated under contractible $G$-space, thereby affirming conjecture Mike Hill.