Realm

Principles

Formal and modeling disciplines that keep distinctions precise across the graph.

Entries

Adjunctions

Adjunctions describe paired constructions that translate between domains in the best available way, even when the translations are not inverses.

Algebras and Coalgebras

Algebras and coalgebras provide complementary ways to model construction and behavior.

Asynchronous Computability Theorem

The asynchronous computability theorem characterizes which distributed tasks can be solved wait-free in an asynchronous read/write system.

CALM Theorem

The CALM theorem, "Consistency as Logical Monotonicity", states that a program has a consistent, coordination-free distributed implementation if and only if it can be expressed in monotonic logic.

Categorical principles provide modeling discipline for the Cohesive System Model. They are not the entry point for ordinary readers, but they help keep distinctions precise when relating semantic dynamics, system structure, operational semantics, and realization substrate.

Compositionality

Compositionality is the principle that complex systems should be understood from parts and the rules by which those parts compose.

Database Sheaf Semantics

Database Sheaf Semantics views a database schema as a category, a database instance as a structure-preserving functor into sets, and local database views as sections that can be restricted, compared on overlaps, and sometimes glued into a coherent larger instance.

Duality and Symmetry

Duality and symmetry are principles for recognizing paired concepts that explain one another through reversal, complementarity, or mirrored structure.

Enrichment and Order

Enrichment adds structure to relationships. Instead of merely asking whether a relationship exists, an enriched view asks what kind of value the relationship has: order, distance, cost, probability, time, authority, confidence, or information.

Equivalence vs Equality

Equivalence and equality should not be confused.

Event-State Duality

Event-state duality is a modeling principle and an instance of duality and symmetry: the relationship between two views of behavior:

Fibrations and Indexed Structure

Fibrations and indexed structure describe situations where each object in a base domain has a category of things lying over it.

Fixed Points and Recursion

Fixed points and recursion describe self-referential definitions, repeated behavior, and systems whose outputs feed future inputs.

Functoriality

Functoriality is the principle that a mapping between domains should preserve the structure that matters: identities, relationships, composition, and change.

Monads Monoids and Duals

Monads, monoids, and their duals provide recurring patterns for sequencing, accumulation, context, observation, and composition.

Naturality

Naturality is the principle that a transformation should be independent of arbitrary representation choices. It should commute with the structure-preserving maps between representations.

Optics and Lenses

Optics are structured ways to focus on, observe, transform, or update part of a larger structure.

Sheaves and Gluing

Sheaves and gluing provide local vocabulary for describing systems of observations: many observers, contexts, processors, time intervals, schemas, views, or execution cuts may each see part of a system, and the model needs to say when those partial views agree enough to form a coherent larger view.