Analogy, Cognitive Architecture and Universal Construction: A Tale of Two Systematicities

Cognitive science recognizes two kinds of systematicity: (1) as the property where certain cognitive capacities imply certain other related cognitive capacities (Fodor and Pylyshyn); and (2) as the principle that analogical mappings based on collections of connected relations are preferred over relations in isolation (Gentner). Whether these kinds of systematicity are two aspects of a deeper property of cognition is hitherto unknown. Here, it is shown that both derive from the formal, category-theoretic notion of universal construction. In conceptual/psychological terms, a universal construction is a form of optimization of cognitive resources: optimizing the re-utilization of common component processes for common task components. Systematic cognitive capacity and the capacity for analogy are hallmarks of human cognition, which suggests that universal constructions (in the category-theoretic sense) are a crucial component of human cognitive architecture.


Remark (Composable)
. An arrow f is said to be composable with an arrow g whenever the codomain of f equals the domain of g. The composition of f with g is the arrow g • f .
Remark (Arrow, morphism, map). Arrow, morphism and map are synonymous, though usage varies and is not strict. Here we use: arrow, generally; morphism to emphasize additional structure, e.g., universal morphism includes object and arrow components; and map to emphasize the action on elements for arrows between sets, e.g., map f : Example (Set). The category Set has sets for objects and total functions for arrows, where the identity arrows are the identity functions and composition is function composition.

Example (Poset).
A partially ordered set (poset) is a set P together with a binary relation ≤ on P , denoted as the pair (P, ≤), that is reflexive (p ≤ p), transitive (p ≤ q ∧ q ≤ r ⇒ p ≤ r) and antisymmetric for all a, b ∈ P . The category Poset has posets for objects and monotonic functions for arrows, with identity arrows and composition as for Set.
Analogy and cognitive architecture 2 Example (Discrete category). A discrete category is a category that has no non-identity arrows. A set A is construed as a discrete category by regarding each element a ∈ A as an object and associating with each a an identity arrow, 1 a : a → a.

Example (Poset as category).
A poset (P, ≤) is a category whose objects are the elements p, q ∈ P with one arrow p → q just in case p ≤ q. Definition (Product). In a category C, a product of objects A and B is an object P , also denoted A × B, together with two arrows p 1 : P → A and p 2 : P → B, conjointly denoted (P, p 1 , p 2 ), such that for every object Z ∈ |C| and every pair of arrows f : Z → A and g : Z → B there is a unique arrow together with two functions (projections) for retrieving the first and second elements of each pair, i.e.,

Example (Cartesian product). A Cartesian product of sets A and B, denoted
A Cartesian product is a product in Set.

Example (Greatest lower bound). In a poset (P, ≤), a lower bound of an element p ∈ P is an element
w ∈ P such that w ≤ p. A greatest lower bound (glb) of elements p, q ∈ P (if it exists) is a lower bound r of p and q such that for every lower bound z of p and q, we have z is a lower bound of r (i.e. r ≤ p ∧ r ≤ q and for all z ∈ P we have z ≤ p ∧ z ≤ q ⇒ z ≤ r). A greatest lower bound is a product in a poset as a category.

Definition (Functor).
A functor from a category C to a category D is a map F : C → D that sends each object A in C to an object F (A) in D, and each arrow f : in D, satisfying the following axioms: for all arrows f : A → B and g : B → C in C; and • identity:

Example (Diagonal functor). A diagonal functor, denoted ∆, sends each object A to its pair (A, A)
and each arrow f : A → B to its pair (f, f ). That is, ∆ :

Example (Product functor).
A product functor, denoted Π, sends each pair of objects (A, B) to their product object A × B and each pair of arrows (f, g) to their product arrow f × g. That is, With the definitions of category and functor, we are ready to define universal construction. A universal construction comes in two varieties: universal morphism and couniversal morphism, which is the dual construction and obtained by reversing the directions of the arrows in the definition of universal morphism.
We just provide the definition of universal morphism.

Definition (Universal morphism).
Given a functor F : A → C and an object Y ∈ |C|, a universal morphism from F to Y is a pair consisting of an object A in A, and a arrow φ in C, denoted (A, φ), such that for every object Z ∈ |A| and every arrow f : F (Z) → Y , there exists a unique arrow u : Z → A, Example (Product as universal morphism). A product (A × B, (p 1 , p 2 )) is a universal morphism, and indicated in commutative diagram

Definition (Universal construction).
A universal construction is either a universal morphism, or a couniversal morphism.

Example (Product as universal construction).
A product is a universal construction.