The Ubiquity of Self-Predicative Universality of Adjoint Functors. Note Quote.

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One of the most important and beautiful notions in category theory is the notion of a pair of adjoint functors. The developers of category theory, Saunders MacLane and Samuel Eilenberg, famously said that categories were defined in order to define functors, and functors were defined in order to define natural transformations. Adjoints were defined more than a decade later by Daniel Kan but the realization of their ubiquity (“Adjoint functors arise everywhere” (MacLane) and their foundational importance has steadily increased over time (Lawvere). Now it would perhaps not be too much of an exaggeration to see categories, functors, and natural transformations as the prelude to defining adjoint functors. The notion of adjoint functors includes all the instances of self-predicative universal mapping properties discussed above. As Steven Awodey (179) put it:

The notion of adjoint functor applies everything that we have learned up to now to unify and subsume all the different universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomenon that is invisible without the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

“The isolation and explication of the notion of adjointness is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas.” (Goldblatt)

How do the ubiquitous and important adjoint functors relate to theme of self- predicative universals? MacLane and Birkhoff succinctly state the idea of the self-predicative universals of category theory and note that adjunctions can be analyzed in terms of those universals. The construction of a new algebraic object will often solve a specific problem in a universal way, in the sense that every other solution of the given problem is obtained from this one by a unique homomorphism. The basic idea of an adjoint functor arises from the analysis of such universals. (MacLane and Birkhoff)

We will use a specific novel treatment of adjunctions (Ellerman) that shows they arise by gluing together in a certain way two universal constructions or self-predicative universals (“semi-adjunctions”). But for illustration, we will stay within the methodological restriction of using examples from partial orders (where adjunctions are called “Galois connections”).

We have been working within the inclusion partial order on the set of subsets ζ(U) of a universe set U. Consider the set of all ordered pairs of subsets <a,b> from the Cartesian product ζ(U) x ζ(U) where the partial order (using the same symbol) is defined by pairwise inclusion. That is, given the two ordered pairs <a’, b’> and <a,b>, we define

<a’,b’> ⊆ <a,b> if a ⊆’  a and b ⊆’  b.

Order-preserving maps can be defined each way between these two partial orders. From ζ(U) to ζ(U) x ζ(U), there is the diagonal map Δ(x) = <x,x>, and from ζ(U) x ζ(U) to ζ(U), there is the meet map ∩(<a,b>)  = a ∩ b. Consider now the following “adjointness relation” between the two partial orders:

Δ(c) ⊆ <a,b> iff c ⊆ ∩ (<a,b>) Adjointness Equivalence

for sets a, b, and c in ζ(U). It has a certain symmetry that can be exploited. If we fix <a,b>, then we have the previous universality condition for the meet of a and b: for any c in ζ(U), c ⊆ a ∩ b iff Δ(c) ⊆ <a,b> Universality Condition for Meet of Sets a and b.

The defining property on elements c of ζ(U) is that Δ(c) ⊆ <a,b>. But using the symmetry, we could fix c and have another universality condition using the reverse inclusion in ζ(U) x ζ(U) as the participation relation: for any <a,b> in ζ(U) x ζ(U), <a,b> ⊇ Δ(c) iff c ⊆ a ∩  b. Universality Condition for Δ(c). Here the defining property on elements <a,b> of ζ(U) x ζ(U) is that “the meet of a and b is a superset of the given set c.” The self-predicative universal for that property is the image of c under the diagonal map Δ(c) = <c,c>, just as the self-predicative universal for the other property defined given <a,b> was the image of <a,b> under the meet map ∩(<a,b>) = a ∩ b.

Thus in this adjoint situation between the two categories ζ(U) and ζ(U) x ζ(U), we have a pair of maps (“adjoint functors”) going each way between the categories such that each element in a category defines a certain property in the other category and the map carries the element to the self-predicative universal for that property.

Δ: ζ(U) → ζ(U) x ζ(U) and ∩: ζ(U) x ζ(U) → ζ(U) Example of Adjoint Functors Between Partial Orders

The notion of a pair of adjoint functors is ubiquitous; it is one of the main tools that highlights self-predicative universals throughout modern mathematics.

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