Grounded theory and a finite colimit
Try a precise, limited analogy for coding incidents. You decide which incidents must receive the same code; the model computes exactly the identifications those decisions imply. It does not infer meaning or discover a core theory.
Illustrative incidents only. Changes stay in this page until reset or reload; nothing is saved or uploaded by these controls.
1. Make your identifications explicit
A link here means “use the same code”, not just “these incidents are related”. Equality is symmetric and transitive, even though the declared map names a source and a target.
| Quotient class | Incidents mapped to it |
|---|
2. Read the coding analogy
3. Test a proposed coding
Give each incident a non-empty code. A compatible assignment must be constant on every quotient class. Different classes may share a code: that adds a further distinction you choose to collapse.
What is actually computed?
The indexing category is the free category on the chosen directed graph. Each incident is represented by a singleton set, and each arrow by its unique singleton-to-singleton map. The colimit in Set is the disjoint union of those singletons, quotiented by the equivalence relation generated by the arrows. Its elements are the connected components of the identification graph, including isolated incidents.
The canonical map q sends each incident to its class. A candidate map f into a set of code labels factors as f = h ∘ q exactly when its labels agree on every generating arrow. In that case h is unique because every quotient class has an incident representative. The checker shows the unique factor map for the labels you entered; it is an example of the general property, not an empirical test of a theory.
A memo, comparison or causal relation is not automatically an equality or a set map. Choosing identifications is an analyst's modelling decision. There is no automatic single Core node, no claim of theoretical saturation, and no guarantee for unseen incidents. An initial algebra requires a separately specified endofunctor and algebra structure; neither is defined here, so this page makes no initial-algebra claim.
Mathematical construction: The Stacks Project, limits and colimits in the category of sets. The grounded-theory connection is an illustrative interpretation, not a theorem about qualitative research.