Skip to main content Accessibility help

Finitary $\mathcal{M}$ -adhesive categories



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.



Hide All
Adámek, J., Herrlich, H. and Strecker, G. (1990) Abstract and Concrete Categories, Wiley.
Baldan, P., Bonchi, F., Corradini, A., Heindel, T. and König, B. (2011) A Lattice-Theoretical Perspective on Adhesive Categories. Journal of Symbolic Computation 46 222245.
Baldan, P., Bonchi, F., Heindel, T. and König, B. (2008) Irreducible Objects and Lattice Homomorphisms in Adhesive Categories. In: Pfalzgraf, J. (ed.) Proceedings of ACCAT '08 (Workshop on Applied and Computational Category Theory).
Braatz, B. (2009) Formal Modelling and Application of Graph Transformations in the Resource Description Framework, Dissertation, Technische Universität Berlin.
Braatz, B. and Brandt, C. (2008) Graph transformations for the Resource Description Framework. In: Ermel, C., Heckel, R. and de Lara, J. (eds.) Proceedings GT-VMT 2008. Electronic Communications of the EASST 10.
Braatz, B., Ehrig, H., Gabriel, K. and Golas, U. (2010) Finitary $\mathcal{M}$-Adhesive Categories. In: Proceedings ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372 234249.
Braatz, B., Golas, U. and Soboll, T. (2011) How to delete categorically – two pushout complement constructions. Journal of Symbolic Computation 46 246271.
Cockett, J. R. B. and Lack, S. (2002) Restriction categories I: categories of partial maps. Theoretical Computer Science 270 (1–2)223259.
Corradini, A., Heindel, T., Hermann, F. and König, B. (2006) Sesqui-Pushout Rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L. and Rozenberg, G. (eds.) Proceedings of ICGT 2006. Springer-Verlag Lecture Notes in Computer Science 4178 3045.
Ehrig, H. (1979) Introduction to the Algebraic Theory of Graph Grammars (A Survey). In: Claus, V., Ehrig, H. and Rozenberg, G. (eds.) Graph Grammars and their Application to Computer Science and Biology. Springer-Verlag Lecture Notes in Computer Science 73 169.
Ehrig, H., Ehrig, K., Prange, U. and Taentzer, G. (2006a) Fundamentals of Algebraic Graph Transformation, EATCS Monographs, Springer-Verlag.
Ehrig, H., Golas, U. and Hermann, F. (2010) Categorical Frameworks for Graph Transformations and HLR Systems based on the DPO Approach. Bulletin of the EATCS 102 111121.
Ehrig, H., Padberg, J., Prange, U. and Habel, A. (2006b) Adhesive High-Level Replacement Systems: A New Categorical Framework for Graph Transformation. Fundamenta Informaticae 74 (1)129.
Habel, A. and Pennemann, K.-H. (2009) Correctness of High-Level Transformation Systems Relative to Nested Conditions. Mathematical Structures in Computer Science 19 (2)245296.
Heckel, R., Ehrig, H., Wolter, U. and Corradini, A. (2001) Double-pullback transitions and coalgebraic loose semantics for graph transformation systems. Applied Categorical Structures 9 (1)83110.
Heindel, T. (2010) Hereditary Pushouts Reconsidered. In: Proceedings ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372 250265.
Klyne, G. and Carroll, J. J. (2004) Resource Description Framework (RDF): Concepts and Abstract Syntax, World Wide Web Consortium (W3C). Available at
Lack, S. and Sobociński, P. (2004) Adhesive Categories. In: Walukiewicz, I. (ed.) Proceedings FOSSACS 2004. Springer-Verlag Lecture Notes in Computer Science 2987 273288.
Lack, S. and Sobociński, P. (2005) Adhesive and Quasiadhesive Categories. Theoretical Informatics and Applications 39 (2)511546.
Löwe, M. (2010) Graph Rewriting in Span-Categories. In: Proceedings ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372 218233.
Löwe, M. and Ehrig, H. (1990) Algebraic Approach to Graph Transformation Based on Single Pushout Derivations. In: Möhrung, R. (ed.) Proceedings Graph-Theoretic Concepts in Computer Science. Springer-Verlag Lecture Notes in Computer Science 484 338353.
MacLane, S. (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag.
Modica, al. (2010) Low- and High-Level Petri Nets with Individual Tokens. Technical Report 2009/13, Technische Universität Berlin. Available at
Prange, U., Ehrig, H. and Lambers, L. (2008) Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories. Applied Categorical Structures 16 (3)365388.
Rozenberg, G. (ed.) (1997) Handbook of Graph Grammars and Computing by Graph Transformation 1: Foundations, World Scientific.

Related content

Powered by UNSILO

Finitary $\mathcal{M}$ -adhesive categories



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.