Finitary $\mathcal{M}$-adhesive categories. (26th June 2014) - GABRIEL, KARSTEN; BRAATZ, BENJAMIN; EHRIG, HARTMUT; GOLAS, ULRIKE
Abstract : Finitary $\mathcal{M}$ -adhesive categories are $\mathcal{M}$ -adhesive categories with finite objects only, where $\mathcal{M}$ -adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of $\mathcal{M}$ -subobjects. In this paper, we show that in finitary $\mathcal{M}$ -adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for $\mathcal{M}$ -adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary $\mathcal{M}$ -adhesive categories have a unique $\mathcal{E}$ - $\mathcal{M}$ factorisation and initial pushouts, and the existence of an $\mathcal{M}$ -initial object implies we also have finite coproducts and a unique $\mathcal{E}$ ′- $\mathcal{M}$ pair factorisation. Moreover, we can show that the finitary restriction of each $\mathcal{M}$ -adhesive category is a finitary $\mathcal{M}$ -adhesive category, and finitarity is preserved under functor and comma category constructions based on $\mathcal{M}$ -adhesive categories. This means that all the classical results are also valid for corresponding finitary $\mathcal{M}$ -adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non- $\mathcal{M}$ -adhesive categories.