Strict initial object

Abstract: We call a nitely complete category diexact if every difunctional relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: rstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with nite limits and pushouts of difunctional relations is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos.

Abstract: We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.

Abstract: We show that every adhesive category gives rise to an associative algebra of rewriting rules induced by the notion of double-pushout (DPO) rewriting and the associated notion of concurrent production. In contrast to the original formulation of rule algebras in terms of relations between (a concrete notion of) graphs, here we work in an abstract categorical setting. Doing this, we extend the classical concurrency theorem of DPO rewriting and show that the composition of DPO rules along abstract dependency relations is, in a natural sense, an associative operation. If in addition the adhesive category possesses a strict initial object, the resulting rule algebra is also unital. We demonstrate that in this setting the canonical representation of the rule algebras is obtainable, which opens the possibility of applying the concept to define and compute the evolution of statistical moments of observables in stochastic DPO rewriting systems.