A lattice-theoretical perspective on adhesive categories

It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows

Structural Decomposition of Reactions of Graph-Like Objects

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation. Introduction Compositional methods for the synthesis and analysis of computational systems remain a fruitful research topic with potential applications in practice. Though compositionality is most clearly exhibited in semantics for process calculi where structural operational semantics (SOS) can be found in its "pure" form, a slightly broader perspective is appropriate to make use of the fundamental ideas of SOS in interdisciplinary research. The first source of inspiration of the present paper is the κ-calculus [6], which is an influential modelling framework in Computational Systems Biology. The κ-calculus allows to give abstract, formal descriptions of biological systems that can be used to explain the reaction (rate) of complex systems, so-called complexes, in terms of the reaction (rate) of each of its subsystems, which are called partial complexes. Leaving quantitative aspects as a topic for future research, we concentrate on a specific sub-problem, namely the "purely structural" decomposition of reactions. In the κ-calculus, system states are composed of partial complexes and they have an intuitive, graphical representation. Hence, it is natural to investigate the decomposition of (reactions of) system states using concepts from graph transformation. In its simplest form, the idea of composition of graph transformations is by means of coproducts. Intuitively, the coproduct of two graphs models the assembly of two states put side by side and the two (sub-)states react independently of each other. A well-known, related theorem about graph transformations is the so-called Parallelism Theorem (see e.g. [5, Theorem 17]). A more general formalism of compositionality that is based on pushouts has been (re-)considered in In this paper, we shall remove the restriction to pushouts as a composition mechanism and generalize the results of [18] from pushouts to (pullback stable) colimits of arbitrary shape. This considerably enlarges the set of available gluing patterns. As a simple example, we can now equip each sub-state with several interfaces; this would be appropriate for the model of a cell in an organism that is in direct contact with each of its neighbouring cells with some part of its membrane; each area of contact would be modelled by a different interface. Content of the paper After reviewing some basic category theoretical concepts and the definition of adhesive categories in Section 1, we begin Section 2 with the "deconstruction" of models of system states; more precisely, we explain in Section 2.1 how suitably finite objects in adhesive categories arise as the colimit of a diagram of "atomic" objects, namely irreducible objects in the sense of The main problem, which is concerned with the decomposition of a "global" transformation into a family of "local" ones, is addressed in Section 3. We give a formal description of local decomposition problems, which consist of a given decomposition of a state (as a colimit of a certain shape) and a rule that describes a possible reaction of the state; to solve such a problem means to extend the decomposition of the state to a decomposition of the whole reaction (using colimits of the same shape). Section 3.1 presents a "global" solution, which first constructs the whole transformation "globally"; a "more local" solution of the problem is possible if we are given extra information that involve a generalization of the accommodations o