Categorical Frameworks for Graph Transformation and HLR Systems based on the DPO Approach

Several variants of high-level replacement (HLR) and adhesive categories have been introduced in the literature as categorical frameworks for graph transformation and HLR systems based on the double pushout (DPO) approach. In addition to HLR, adhesive, and adhesive HLR categories several weak variants, especially weak adhesive HLR with horizontal and vertical variants, as well as partial variants, including partial map adhesive and partial VK square adhesive categories are reviewed and related to each other. We propose as weakest version the class of vertical weak adhesive HLR categories, short $\mathcalM$-adhesive categories, which are still sufficient to obtain most of the main results for graph transformation and HLR systems. The results in this paper are summarized in Fig.~f:hierarchy showing a hierarchy of all these variants of adhesive, adhesive HLR, and $\mathcalM$-adhesive categories, which can be considered as different categorical frameworks for graph transformation and HLR systems.