Shared watersCorrected attempt 01 / 03
Two diagrams.
One shared interface.
A depot needs brief informs a draft barge service offer, which informs a receiving plan. Each local graph has an inside and a named boundary. Composition identifies the matching handover; it does not invent another task between the two.
Watch the boundary become shared.
Circles are interface vertices. Arrows are declared transformations of service information, not power flows or elapsed time. The ochre interface is exposed by both neighbours. The composed graph keeps the drafting and planning steps.
The thing being learned
Gluing means identifying.
Before the join, each module has its own copy of the draft-offer interface. We declare that the two copies represent the same boundary. After the join, there is one shared vertex.
The input and output of the whole remain exposed. Its internal handover still exists in the graph; it has become internal to the composite.
With three matching modules, the two ways of grouping the joins produce the same graph up to renaming. The check compares the finite graph construction, not every possible open-system theory.
A useful distinction
A matching name alone is not enough. This example declares the boundary identity and its type. A different interface is rejected.
Matching the graph boundary still does not establish staffing, compatible histories or a real agreement. Those would need an interpretation and further conditions.
compose through the common boundary B.
Here A, B and C are discrete boundary graphs included in directed graphs G and H. The arrows in this formula are boundary maps, not task arrows.
The second idea in the original
Changing a view is a different operation.
The first drawing also asks how one whole graph can map to another. Here two parallel ways to prepare a draft offer are both represented by one abstract “prepare” edge. Endpoints and boundary types are preserved; the choice of route is forgotten.
This is a graph map. It changes the description, not the organisation. It does not prove that the two preparation routes take the same time, use the same resources or can substitute for one another.
From the 2022 drawing
What survives the correction?
2022 07 Open Graphs as a double category.graphml
Retained
Open graphs compose through explicit boundaries; a whole-model map must account for its boundaries too.
Corrected
“Glue together” becomes a declared identification of typed interface vertices. The representation map has explicit vertex and edge assignments.
Left out of this attempt: The incomplete double-category square, unsupported 2-cell/interchange claims and empty “3” placeholder. No double category is claimed here.
Source → concept → construction → limit
Willems (2007), printed pp.46–54 and60–64, motivates tearing a system into local models and linking declared interfaces. Baez, part3, distinguishes composition syntax from its interpretation.
The typed finite-graph gluing and graph-map example here are explicit elementary constructions. They are not a claim that the original drawing already supplied a behavioral model or double category. Graph-map endpoint preservation is weaker than preservation of all possible behaviour.
Original GraphML anchors: n0::n0::n0; n0::n0::n1::n0. These identify the local source drawing; they are not mathematical claims. Original files remain unchanged. This site depicts the corrected construction.