Abstract
Colimits are a powerful tool for the combination of objects in a category. In the context of modeling and specification, they are used in the institution-independent semantics (1) of instantiations of parameterised specifications (e.g. in the specification language CASL), and (2) of combinations of networks of specifications (in the OMG standardised language DOL).
The problem of using colimits as the semantics of certain language constructs is that they are defined only up to isomorphism. However, the semantics of a complex specification in these languages is given by a signature and a class of models over that signature – not by an isomorphism class of signatures. This is particularly relevant when a specification with colimit semantics is further translated or refined. The user needs to know the symbols of a signature for writing a correct refinement.
Therefore, we study how to usefully choose one representative of the isomorphism class of all colimits of a given diagram. We develop criteria that colimit selections should meet. We work over arbitrary inclusive categories, but start the study how the criteria can be met with \(\mathbb Set\)-like categories, which are often used as signature categories for institutions.
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Notes
- 1.
When we use the term specification, our theory applies equally to ontologies and models, provided these have a formal semantics as theories of some institution.
- 2.
That is, with the same objects as \(\mathbf {C}\).
- 3.
For inclusive categories for which the symbol functor uniquely lifts colimits, solving colimit selection for \(\mathbb Set\) already suffices. However, the inclusive categories studied in Sect. 4.3 typically do not enjoy this property.
- 4.
It is straightforward but not essential here to make the notion of sentence precise.
- 5.
CASL has a mechanism of “compound identifiers” that ensures name-clash-freeness in multiple instantiations of parametrised specifications, such as List[List[Elem]], see [16], p.47f. and p.224f.
- 6.
Note that this construction extends to institutions, yielding Grothendieck institutions, see [5].
- 7.
In some languages, # is used instead of/. But this has the disadvantage that, when used as an IRL, the fragment following the # is not transmitted to servers.
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Mossakowski, T., Rabe, F., Codescu, M. (2017). Canonical Selection of Colimits. In: James, P., Roggenbach, M. (eds) Recent Trends in Algebraic Development Techniques. WADT 2016. Lecture Notes in Computer Science(), vol 10644. Springer, Cham. https://doi.org/10.1007/978-3-319-72044-9_12
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