Some fundamental algebraic tools for the semantics of computation, part 3: indexed categories
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A distributed graph (N,D) consists of a network graph N and a commutative diagram D over the scheme N which associates local graphs D(ni) and graph morphisms D(e): D(n1) →D(n2) to nodes n1, n2 and edges e: n1 →n2 in N. Although there are several interesting applications of distributed graphs and transformations, even the basic pushout constructions for the double pushout approach of distributed graph transformation could be shown up to now only in very special cases. In this paper we show that the category of distributed graphs can be considered as a Grothendieck category over a specific indexed category, which assigns to each network N the category of all diagrams D of shape N. In this framework it is possible to give a free construction which allows to construct for each diagram D1 over N1 and network morphism h:N1 →N2 a free extension Fh(D1) over N2 and to show that the Grothendieck category is complete and cocomplete if the underlying category of local graphs has these properties. Moreover, an explicit construction for general pushouts of distributed graphs is given. This pushout construction is based on the free construction. The non-trivial proofs for free constructions and pushouts are the main contributions of this paper and they are compared with the special cases known up to now.