Reprints in Theory and Applications of Categories, No. 16, 2006, pp. 1–16.
ADJOINTNESS IN FOUNDATIONS
F. WILLIAM LAWVERE
Author’s commentary
In this article we see how already in 1967 category theory had made explicit a number of
conceptual advances that were entering into the everyday practice of mathematics. For
example, local Galois connections (in algebraic geometry, model theory, linear algebra,
etc.) are globalized into functors, such as Spec, carrying much more information. Also,
“theories” (even when presented symbolically) are viewed explicitly as categories; so are
the background universes of sets that serve as the recipients for models. (Models them-
selves are functors, hence preserve the fundamental operation of substitution/composition
in terms of which the other logical operations can be characterized as local adjoints.)
My 1963 observation (referred to by Eilenb erg and Kelly in La Jolla, 1965), that carte-
sian closed categories serve as a common abstraction of type theory and propositional
logic, permits an invariant algebraic treatment of the essential problem of proof theory,
though most of the later work by pro of theorists still relies on presentation-dependent
formulations. This article sums up a stage of the development of the relationship between
category theory and proof theory. (For more details see Proceedings of the AMS Sympo-
sium on Pure Mathematics XVII (1970), pp. 1–14, and Marcel Dekker, Lecture Notes in
Pure and Applied Mathematics, no. 180 (1996), pp. 181–189.)
The main problem addressed by proof theory arises from the existential quantifier in
“there exists a proof. . . ”. The strategy to interpret proofs themselves as structures had
been discussed by Kreisel; however, the influential “realizers” of Kleene are not yet the
usual mathematical sort of structures. Inspired by L¨auchli’s 1967 success in finding a
completeness theorem for Heyting predicate calculus lurking in the category of ordinary
permutations, I presented, at the 1967 AMS Los Angeles Symposium on Set Theory, a
common functorization of several geometrical structures, including such proof-theoretic
structures. As Hyperdoctrines, those structures are described in the Proceedings of the
Author’s commentary copyright F. William Lawvere
Originally published: Dialectica, 23 (1969), used by permission.
My heartfelt thanks go to Christoph Schubert who expertly rendered the original article into T
E
X
and to the editors of TAC for their work and dedication.
Received by the editors of TAC February 22, 2006 and by Dialectica December 15, 1967.
Transmitted by R. Par´e, R. Rosebrugh and R.J. Wood. Republished on 2006-05-21.
2000 Mathematics Subject Classification: 00A30, 14F20, 18B25, 18B30, 18B40, 55U40, 81P05.
Key words and phrases: Formal-conceptual duality, cartesian-closed categories, algebraic logic,
globalized Galois connections.
1
2 F. WILLIAM LAWVERE
AMS New York Symposium XVII, cited above.
Proof theory may be regarded as the study of the presentations of certain algebraic
structures, for example, locally cartesian closed categories C with finite coproducts. The
map category C/X models “proof bundles” with its morphisms playing the role of de-
ductions. Then PX, defined as the poset reflection of C/X, is a Heyting algebra; any
map X → Y in C contravariantly induces a Heyting homomorphism of substitution and
covariantly induces existential and universal quantifier operations that not only satisfy
the correct rules of inference but, moreover, satisfy the usual proof-theoretic slogans that
(a) a proof of an existential statement includes the specification of an element of the
kind required by the statement;
(b) a proof of a universal statement is a functional giving a uniform proof of all the
instances.
The poset reflection expresses the idea of “there exists a deduction A → B” in C/X, and
PX serves as a system of “proof-theoretic propositions” about elements of X. In case C
is actually a topos, there is a natural map
PX → P
C
(X)
to the usual subobject lattice, defined by taking the image of any A → X. T his map will
be an isomorphism for all X only if C satisfies the axiom of choice, but we might hope
that at least P is “small” like P
C
and that the idempotent of P whose splitting is P
C
could be described in a useful way. There are two results about that.
(1) The Heyting algebras PX are small iff C itself is B oolean, as in L¨auchli’s original
construction.
(2) If we localize the definition of P in the sense that we do not ask for the existence of
maps A → B in C/X, but only for maps A
0
→ B where A
0
has an epimorphism to
A, then this coarsened version of P is actually isomorphic to P
C
for all toposes C.
The second result divides the existential quantifier in the proof problem into two steps;
the preliminary search for an appropriate A
0
which covers A illustrates that a hypothesis
must sometimes be analyzed before it can be used to launch a deduction. The collapse
from a type system to the corresponding system of proof-theoretic propositions is of
course not an “isomorphism” (contrary to a fashionable colloquialism), even though both
involve cartesian closed categories. To arrive at mathematical truth, we require the further
collapse resulting from the unbounded preliminary step.
Buffalo, 22 February 2006
ADJOINTNESS IN FOUNDATIONS 3
1. The Formal–Conceptual Duality in Mathematics and in its Foundations
1
That pursuit of exact knowledge which we call mathematics seems to involve in an es-
sential way two dual aspects, which we may call the Formal and the Conceptual. For
example, we manipulate algebraically a polynomial equation and visualize geometrically
the corresponding curve. Or we concentrate in one moment on the deduction of theorems
from the axioms of group theory, and in the next consider the classes of actual groups to
which the theorems refer. Thus the Conceptual is in a certain sense the subject matter
of the Formal.
Foundations will mean here the study of what is universal in mathematics. Thus
Foundations in this sense cannot be identified with any “starting-point” or “justification”
for mathematics, though partial results in these directions may be among its fruits. But
among the other fruits of Foundations so defined would presumably be guide-lines for
passing from one branch of mathematics to another and for gauging to some extent which
directions of research are likely to be relevant.
Being itself part of Mathematics, Foundations also partakes of the Formal-Conceptual
duality. In its formal aspect, Foundations has often concentrated on the formal side
of mathematics, giving rise to Logic. More recently, the search for universals has also
taken a conceptual turn in the form of Category Theory, which began by viewing as a
new mathematical object the totality of all morphisms of the mathematical objects of
a given species A, and then recognizing that these new mathematical objects all belong
to a common non-trivial species C which is independent of A. Naturally, the formal
tendency in Foundations can also deal with the conceptual aspect of mathematics, as
when the semantics of a formalized theory T is viewed itself as another formalized theory
T
0
, or in a somewhat different way, as in attempts to formalize the study of the category
of categories. On the other hand, Foundations may conceptualize the formal aspect of
mathematics, leading to Boolean algebras, cylindric and polyadic algebras, and to certain
of the structures discussed below.
One of the aims of this paper is to give evidence for the universality of the concept
of adjointness, which was first isolated and named in the conceptual sphere of category
theory, but which also seems to pervade logic. Specifically, we describe in section III
the notion of cartesian closed category, which appears to be the appropriate abstract
structure for making explicit the known analogy (sometimes exploited in proof theory)
between the theory of functionality and propositional logic. The structure of a cartesian
closed category is entirely given by adjointness, as is the structure of a “hyperdoctrine”,
which includes quantification as well. Precisely analogous “quantifiers” occur in realms of
mathematics normally considered far removed from the province of logic or proof theory.
1
I inserted the following subtitles into the text for this reprinting:
1. The formal-conceptual duality in mathematics and in its foundations
2. Adjointness in the meta-category of categories
3. Cartesian-closed categories and hyperdoctrines
4. Globalized Galois connections in algebraic geometry and in foundations
4 F. WILLIAM LAWVERE
As we point out, recursion (at least on the natural numbers) is also characterized entirely
by an appropriate adjoint; thus it is possible to give a theory, roughly proof theory
of intuitionistic higher-order number theory, in which all important axioms (logical or
mathematical) express instances of the notion of adjointness.
The above-discussed notions of Conceptual, Formal, and Foundations play no mathe-
matical role in this paper; they were included in this introductory section only to provide
one possible perspective from which to view the relationship of category theory in general
and of this paper in particular to other work of universal tendency. However, if one wished
to take these notions seriously, it would seem to follow that an essential feature of any at-
tempt to f ormalize Foundations would be a description of this claimed “duality” between
the Formal and the Conceptual; indeed, both category theory and set theory succeed to
some extent in providing such a description in certain cases. Concerning the latter point
there is a remark at the end of section IV, which is otherwise devoted to a discussion of
a class of adjoint situations differing from those of section III but likewise seeming to be
of universal significance — these may be described briefly as a sort of globalized Galois
connection.
2. Adjointness in the Meta-Category of Categories
The formalism of category theory is itself often presented in “geometric” terms. In fact,
to give a category is to give a meaning to the word morphism and to the commutativity
of diagrams like
A
f
−−−→ B
A
f
//
h
@
@
@
@
@
@
@
B
g
C
A
f
//
a
B
b
A
0
f
0
//
B
0
etc.
which involve morphisms, in such a way that the obvious associativity and identity con-
ditions hold, as well as the condition that whenever
A
f
−−−→ B and B
g
−−→ C
are commutative then there is just one h such that
A
f
//
h
@
@
@
@
@
@
@
B
g
C
is commutative.
To save printing space, one also says that A is the domain, and B the codomain of f
when
A
f
−−−→ B
ADJOINTNESS IN FOUNDATIONS 5
is commutative, and in particular that h is the composition f.g if
A
f
//
h
@
@
@
@
@
@
@
B
g
C
is commutative. We regard objects as co-extensive with identity morphisms, or equiva-
lently with those morphisms which appear as domains or codomains. As usual we call a
morphism which has a two-side d inverse an isomorphism.
With any category A is associated another A
op
obtained by maintaining the inter-
pretation of “morphism” but reversing the direction of all arrows when re-interpreting
diagrams and their commutativity. Thus A
f
−−−→ B in A
op
means B
f
−−−→ A in A, and
h = f.g in A
op
means h = g.f in A; clearly the interpretations of “domain”, “codomain”
and “composition” determine the interpretation of general “commutativity of diagrams”.
The category-theorist owes an apology to the philosophical reader for this unfortunately
well-established use of the word “commutativity” in a context more general than any
which could reasonably have a description in terms of “interchangeability”.
A functor F involves a domain category A, a codomain category B, and a mapping
assigning to every morphism x in A a morphism xF in B in a fashion which preserves
the commutativity of diagrams. (Thus in particular a functor preserves objects, do-
mains, codomains, compositions, and isomorphisms.) Utilizing the usual composition of
mappings to define commutative diagrams of functors, one sees that functors are the
morphisms of a super-category, usually called the (meta-) category of categories.
A category with exactly one morphism will be denoted by 1. It is determined uniquely
up to isomorphism by the fact that for any category A, there is exactly one functor
A −−→ 1; the functors 1
A
−−−→ A correspond bijectively to the objects in A.
A natural transformation ϕ involves a domain category A, a codomain category B,
domain and codomain functors
A
F
0
//
F
1
//
B
and a mapping assigning to each object A of A a morphism
AF
0
Aϕ
−−−−→ AF
1
in B, in such a way that for every morphism
A
a
−−→ A
0
in A, the diagram
AF
0
Aϕ
//
aF
0
AF
1
aF
1
A
0
F
0
A
0
ϕ
//
A
0
F
1