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Category Theory
Category theory has come to occupy a central position in contemporary
mathematics and theoretical computer science, and is also applied to
mathematical physics. Roughly, it is a general mathematical theory of
structures and of systems of structures. As category theory is still
evolving, its functions are correspondingly developing, expanding and
multiplying. At minimum, it is a powerful language, or conceptual
framework, allowing us to see the universal components of a family of
structures of a given kind, and how structures of different kinds are
interrelated. Category theory is both an interesting object of
philosophical study, and a potentially powerful formal tool for
philosophical investigations of concepts such as space, system, and
even truth. It can be applied to the study of logical systems in which
case category theory is called "categorical doctrines" at the
syntactic, prooftheoretic, and semantic levels. Category theory is an
alternative to set theory as a foundation for mathematics. As such, it
raises many issues about mathematical ontology and epistemology.
Category theory thus affords philosophers and logicians much to use
and reflect upon.
1.1 Definitions
Categories are algebraic structures with many complementary natures,
e.g., geometric, logical, computational, combinatorial, just as groups
are manyfaceted algebraic structures. Eilenberg & Mac Lane
(1945) introduced categories in a purely auxiliary fashion, only as a
formal ground for what they called functors and natural
transformations, recognizing from the outset that categories are wholly
dispensable in practice. The very definition of a category evolved over
time, according to the author's chosen goals and metamathematical
framework. Eilenberg & Mac Lane at first gave a purely
abstract definition of a category, along the lines of the axiomatic
definition of a group. Others, starting with Grothendieck (1957) and
Freyd (1964), elected for reasons of practicality to define categories
in settheoretic terms.
An alternative approach, that of Lawvere (1963, 1966), begins by
characterizing the category of categories, and then stipulates that a
category is an object of that universe. This approach, under active
development by various mathematicians, logicians and mathematical
physicists, lead to what are now called "higherdimensional
categories" (Baez 1997, Baez & Dolan 1998a, Batanin 1998, Leinster
2002, Hermida et al. 2000, 2001, 2002). The very definition
of a category is not without philosophical importance, since one of
the objections to category theory as a foundational framework is the
claim that since categories are defined as sets, category
theory cannot provide a philosophically enlightening foundation for
mathematics. We will briefly go over some of these definitions,
starting with Eilenberg's & Mac Lane's (1945) algebraic
definition. However, before going any further, the following
definition will be required.
Definition: A mapping e will be called an
identity if and only if the existence of any product
eα or βe implies that eα =
α and βe = β
Definition (Eilenberg & MacLane 1945): A
category C is an aggregate Ob of
abstract elements, called the objects of C,
and abstract elements Map, called mappings of
the category. The mappings are subject to the following five
axioms:
(C1) Given three mappings α_{1},
α_{2} and α_{3}, the triple product
α_{3}(α_{2}α_{1}) is defined
if and only if
(α_{3}α_{2})α_{1} is defined.
When either is defined, the associative law
α_{3}(α_{2}α_{1})
=
(α_{3}α_{2})α_{1}
holds. This triple product is written
α_{3}α_{2}α_{1}.
(C2) The triple product
α_{3}α_{2}α_{1} is defined
whenever both products α_{3}α_{2} and
α_{2}α_{1} are defined.
(C3) For each mapping α, there is at least one identity
e_{1} such that αe_{1} is
defined, and at least one identity e_{2} such that
e_{2}α is defined.
(C4) The mapping e_{X} corresponding to each object
X is an identity.
(C5) For each identity e there is a unique object
X of C such that e_{X} =
e.
As Eilenberg & Mac Lane promptly remark, objects play
a secondary role and could be entirely omitted from the definition.
Doing so, however, would make the manipulation of the applications less
convenient. It is practically suitable,and perhaps psychologically more simple
to think in terms of mappings and objects. The term "aggregate" is used
by Eilenberg & Mac Lane themselves, presumably so as to
remain neutral with respect to the background set theory one wants to
adopt.
Eilenberg & Mac Lane defined categories in 1945 for
reasons of rigor. They did not need categories as such, but something
like categories had to be defined so that the notions of functors and
natural transformations — what they were really interested in and using — could be given an explicit and rigorous
presentation.
Things changed in the following ten years, when categories started
to be used in homology theory and homological algebra. Mac Lane,
Buchsbaum, Grothendieck and Heller were considering categories in which
the collections of morphisms between two fixed objects have an additional
structure. More specifically, given any two objects X and
Y of a category C, the set
Hom(X, Y) of morphisms from
X to Y form an abelian group. Furthermore, for
reasons related to the ways homology and cohomology theories are
linked, the definition of a category had to satisfy an additional
formal property (which we will leave aside for the moment): it had to
be selfdual. These requirements lead to the following definition.
Definition: A category C can be
described as a set Ob, whose members are the objects
of C, satisfying the following three conditions:
Morphism : For every pair X,
Y of objects, there is a set Hom(X,
Y), called the morphisms from X to
Y in C. If f is a
morphism from X to Y, we write
f : X → Y.
Identity : For every object X, there exists
a morphism id_{X} in
Hom(X, X), called the
identity on X.
Composition : For every triple X,
Y and Z of objects, there exists a partial binary
operation from Hom(X, Y) ×
Hom(Y, Z) to
Hom(X, Z), called the composition of
morphisms in C. If
f : X → Y and
g : Y → Z, the
composition of f and g is notated
(g
f ) : X → Z.
Identity, morphisms, and composition satisfy two axioms:
Associativity : If
f : X → Y,
g : Y → Z and
h : Z → W, then
h
(g
f)
= (h
g)
f.
Identity : If f :
X → Y, then (id_{Y}
f) = f and
(f
id_{X}) = f.
This is the definition one finds in most textbooks of category
theory. As such it explicitly relies on a set theoretical background
and language. An alterative, suggested by Lawvere in the early
sixties, is to develop an adequate language and background framework
for a category of categories. We will not present the formal framework
here, for it would take us too far from our main concern, but the
basic idea is to define what are called weak ncategories
(and weak ωcategories), and what had been called categories
would then be called weak 1categories (and sets would be weak
0categories). (See, for instance, Baez 1997 or Makkai 1998.)
Also in the sixties, Lambek proposed to look at categories as
deductive systems. This begins with the notion of a graph,
consisting of two classes Arrows and Objects, and
two mappings between them, s :
Arrows → Objects and
t : Arrows → Objects,
namely the source and the target mappings. The arrows are usually
called the "oriented edges" and the objects "nodes" or
"vertices". Following this, a deductive system is a graph
with a specified arrow:
(R1) id_{X} : X
→ X,
and a binary operation on arrows:
(R2) Given f : X
→ Y and g : Y →
Z, the composition of f and
g is (g
f) : X →
Z.
Of course, the objects of a deductive system are normally thought of
as formulas, the arrows are thought of as proofs or
deductions, and operations on arrows are thought of as
rules of inference. A category is then defined
thus:
Definition (Lambek): A category is a
deductive system in which the following equations hold between proofs:
for all f : X → Y,
g : Y → Z and
h: Z → W,
(E1) f
id_{X} =
f, id_{Y}
f =
f, h
(g
f) = (h
g)
f.
Thus, by imposing an adequate equivalence relation upon proofs, any
deductive system can be turned into a category. It is therefore
legitimate to think of a category as an algebraic encoding of a
deductive system. This phenomenon is already wellknown to logicians,
but probably not to its fullest extent. An example of such an
algebraic encoding is the LindenbaumTarski algebra, a Boolean algebra
corresponding to classical propositional logic. Since a Boolean
algebra is a poset, it is also a category. (Notice also that Boolean
algebras with appropriate homomorphisms between them form another
useful category in logic.) Thus far we have merely a change of
vocabulary. Things become more interesting when firstorder and
higherorder logics are considered. The LindenbaumTarski algebra for
these systems, when properly carried out, yields categories, sometimes
called "conceptual categories" or "syntactic categories"
(Mac Lane & Moerdijk 1992, Makkai & Reyes 1977, Pitts
2000).
1.2 Examples
Almost every known example of a mathematical structure with the
appropriate structurepreserving map yields a category.
 The category Set with objects sets and morphisms the usual
functions. There are variants here: one can consider partial functions
instead, or injective functions or again surjective functions. In each
case, the category thus constructed is different
 The category Top with objects topological spaces and
morphisms continuous functions. Again, one could restrict morphisms to
open continuous functions and obtain a different
category.
 The category hoTop with objects topological spaces and
morphisms equivalence classes of homotopic functions. This category is
not only important in mathematical practice, it is at the core of
algebraic topology, but it is also a fundamental example of a category
in which morphisms are not structure preserving
functions.
 The category Vec with objects vector spaces and morphisms
linear maps.
 The category Diff with objects differential manifolds and
morphisms smooth maps.
 The categories Pord and PoSet with objects
preorders and posets, respectively, and morphisms monotone
functions.
 The categories Lat and Bool with objects lattices
and Boolean algebras, respectively, and morphisms structure preserving
homomorphisms, i.e.
(, ⊥,
,
) homomorphisms.
 The category Heyt with objects Heyting algebras and
(, ⊥,
,
, → )
homomorphisms.
 The category Mon with objects monoids and morphisms monoid
homomorphisms.
 The category AbGrp with objects abelian groups and
morphisms group homomorphisms, i.e. (1, ×, ?) homomorphisms
 The category Grp with objects groups and morphisms group
homomorphisms, i.e. (1, ×, ?) homomorphisms
 The category Rings with objects rings (with unit) and
morphisms ring homomorphisms, i.e. (0, 1, +, ×)
homomorphisms.
 The category Fields with objects fields and morphisms
fields homomorphisms, i.e. (0, 1, +, ×) homomorphisms.
 Any deductive system T with objects formulae and morphisms
proofs.
These examples nicely illustrates how category theory treats the
notion of structure in a uniform manner. Note that a category is characterized by its
morphisms, and not by its objects. Thus the category of topological spaces
with open maps differs from the category of topological spaces with
continuous maps  or, more to the point, the categorical properties of the
latter differ from those of the former.
We should underline again the fact that not all categories are made
of structured sets with structurepreserving maps. Thus any preordered
set is a category. For given two elements p, q of a
preordered set, there is a morphism
f : p → q if and only if
p ≤ q. Hence a preordered set is a category in which there
is at most one morphism between any two objects. Any monoid (and thus
any group) can be seen as a category: in this case the category has
only one object, and its morphisms are the elements of the monoid.
Composition of morphisms corresponds to multiplication of monoid
elements. That the monoid axioms correspond to the category axioms is
easily verified.
Hence the notion of category generalizes those of preorder and
monoid. We should also point out that a groupoid has a very simple
definition in a categorical context: it is a category in which every
morphism is an isomorphism, that is for any morphism
f : X → Y, there is a
morphism g : Y → X such that
f
g =
id_{X} and
g
f =
id_{Y}.
1.3 Fundamental Concepts of the Theory
Category theory unifies mathematical structures in two different ways.
First, as we have seen, almost every set theoretically defined
mathematical structure with the appropriate notion of homomorphism
yields a category. This is a unification provided within a set
theoretical environment. Second, and perhaps even more important, once
a type of structure has been defined, it is imperative to determine how
new structures can be constructed out of the given one. For instance,
given two sets A and B, set theory allows us to
construct their Cartesian product A × B. It is also
imperative to determine how given structures can be decomposed into
more elementary substructures. For example, given a finite Abelian
group, how can it be decomposed into a product of certain of its
subgroups? In both cases, it is necessary to know how structures of a
certain kind may combine. The nature of these combinations might appear
to be considerably different when looked at from a purely set
theoretical perspective.
Category theory reveals that many of these constructions are in fact
certain objects in a category having a "universal property". Indeed,
from a categorical point of view, a Cartesian product in set theory, a
direct product of groups (Abelian or otherwise), a product of
topological spaces, and a conjunction of propositions in a deductive
system are all instances of a categorical product characterized by a
universal property. Formally, a product of two objects
X and Y in a category C is an object
Z of C together with two morphisms,
called the projections, p : Z
→ X and q : Z →
Y such that—and this is the universal property—for all
objects W with morphisms f :
W → X and g : W
→ Y, there is a unique morphism
h : W → Z such that
p
h = f and q
h =
g.
Note that we have defined a product for X and
Y and not the product for X and Y.
Indeed, products and other objects with a universal property are
defined only up to a (unique) isomorphism. Thus in category theory, the
nature of the elements constituting a certain construction is
irrelevant. What matters is the way an object is related to the other
objects of the category, that is, the morphisms going in and the
morphisms going out, or, put differently, how certain structures can be
mapped into a given object and how a given object can map its structure into other structures
of the same kind.
Category theory reveals how different kinds of structures are
related to one another. For instance, in algebraic topology,
topological spaces are related to groups (and modules, rings, etc.) in
various ways (such as homology, cohomology, homotopy, Ktheory). As noted
above, groups with group homomorphisms constitute a category. Eilenberg
& Mac Lane invented category theory precisely in order to
clarify and compare these connections. What matters are the morphisms
between categories, given by functors. Informally, morphisms are
structurepreserving maps between categories. Given two categories
C and D, a functor F from
C to D sends objects of
C to objects of D, and morphisms of
C to morphisms of D, in such a way
that composition of morphisms in C is preserved, i.e.,
F(g
f) =
F(g)
F(f),
and identity morphisms are preserved, i.e.,
F(id_{X}) =
id_{FX}. It immediately follows that
a functor preserves commutativity of diagrams between categories.
Homology, cohomology, homotopy, Ktheory are all example of
functors.
A more direct example is provided by the power set operation, which
yields two functors on the category of sets, depending on how one
defines its action on functions. Thus given a set X,
℘(X) is the usual set of subsets of X, and
given a function f : X →
Y, ℘(f) :
℘(X) → ℘(Y) takes a subset
A of X and maps it to B =
f(A), the image of
f restricted to A in X. It
is easily verified that this defines a functor from the category of
sets into itself.
In general, there are many functors between two given categories, and
the question of how they are connected suggests itself. For instance, given a
category C, there is always the identity functor from
C to C which sends every
object/morphism of C to itself. In particular, there
is the identity functor over the category of sets.
Now, the identity functor is related in a natural manner to the power set functor
described above. Indeed, given a set X and
its power set ℘(X), there is a function
h_{X} which takes an element
x of X and sends it to the singleton set {x},
a subset of X, i.e., an element of ℘(X). This
function in fact belongs to a family of functions indexed by the
objects of the category of sets
{h_{Y} : Y →
℘(X)  Y in Ob(Set)}.
Moreover, it satisfies the following commutativity condition: given any
function f : X → Y, the
identity functor yields the same function
Id(f) :
Id(X) →
Id(Y). The commutativity condition
thus becomes: h_{Y}
Id(f) =
℘(f)
h_{X}.
Thus the family of functions h() relates the
two functors in a natural manner. Such families of morphisms are called
natural transformations between functors. Similarly, natural
transformations between models of a theory yield the usual
homomorphisms of structures in the traditional set theoretical
framework.
The above notions, while important, are not fundamental to category
theory. The latter heading arguably include the notions of
limit/colimit; in turn, these are special cases of what is certainly
the cornerstone of category theory, the concept of adjoint functors,
first defined by Daniel Kan in 1956 and published in 1958.
Adjoint functors can be thought of as being conceptual inverses.
This is probably best illustrated by an example. Let
U : Grp → Set be the forgetful
functor, that is, the functor that sends to each group G its
underlying set of elements U(G), and to a group
homomorphism f : G → H
the underlying set function
U(f) : U(G) →
U(H). In other words, U forgets about the
group structure and forgets the fact that morphisms are group
homomorphisms. The categories Grp and
Set are certainly not isomorphic, as categories, to
one another. (A simple argument runs as follows: the category
Grp has a zero object, whereas Set
does not.) Thus, we certainly cannot find an inverse, in the usual
algebraic sense, to the functor U. But there are many
nonisomorphic ways to define a group structure on a given set
X, and one might hope that among these constructions at least
one is functorial and systematically related to the functor U.
What is the conceptual inverse to the operation of forgetting all the
group theoretical structure and obtaining a set? It is to construct a
group from a set solely on the basis of the concept of group and
nothing else, i.e., with no extraneous relation or data. Such a group is
constructed "freely"; that is, with no restriction whatsoever except
those imposed by the axioms of the theory. In other words, all that is
remembered in the process of constructing a group from a given set is
the fact that the resulting construction has to be a group. Such a
construction exists; it is functorial and it yields what are called
free groups. In other words, there is a functor
F : Set → Grp,
which to any set X assigns the free group
F(X) on X, and to each function
f : X → Y, the group
homomorphism F(f) :
F(X) → F(Y), defined in the
obvious manner. The situation can be described thusly: we have two
conceptual contexts, a group theoretical context and a set theoretical
context, and two functors moving systematically from one context to the
other in opposite directions. One of these functors is elementary,
namely the forgetful functor U. It is apparently trivial and
uninformative. The other functor is mathematically significant and
important. The surprising fact is that F is related to
U by a simple rule and, in some sense, it arises from
U. One of the striking features of adjoint situations is
precisely the fact that fundamental mathematical and logical
constructions arise out of given and often elementary functors.
The fact that U and F are conceptual inverses
expresses itself formally as follows: applying F first and
then U does not yield the original set X, but there
is a fundamental relationship between X and
UF(X). Indeed, there is a function η :
X → UF(X), called the unit of the
adjunction, that simply sends each element of X to itself
in UF(X) and this function satisfies the following
universal property: given any function
g : X →
U(G), there is a unique group homomorphism
h : F(X) →
G such that U(h)
η =
g. In other words, UF(X) is
the best possible solution to the problem of inserting elements of
X into a group (what is called "insertion of generators" in
the mathematical jargon). Composing U and F in the
opposite order, we get a morphism ξ : FU(G)
→ G, called the counit of the adjunction,
satisfying the following universal property: for any group homomorphism
g : F(X) →
G, there is a unique function
h : X → U(G)
such that ξ
F(h) = g
FU(G)
constitutes the best possible solution to the problem of finding a
representation of G as a quotient of a free group. If
U and F were simple algebraic inverses to one
another, we would have the following identity: UF =
I_{Set} and FU =
I_{Grp}, where
I_{Set} denotes the identity functor
on Set and I_{Grp}
the identity functor on Grp. As we have indicated,
these identities certainly do not hold in this case. However, some
identities do hold: they are best expressed with the help of the
commutative diagrams:
U 
η
U → 
UFU 

F 
F
η → 
FUF 

↘ 
↓U
η 


↘ 
↓ξ
F 


U 



F 
where the diagonal arrows denote the appropriate identity natural
transformations.
This is but one case of a very common situation: every free
construction can be described as arising from an appropriate forgetful
functor between two adequately chosen categories. The number of
mathematical constructions that can be described as adjoints is simply
stunning. Although the details of each one of these constructions vary
considerably, the fact that they can all be described using the same
language illustrates the profound unity of mathematical concepts and
mathematical thinking. Before we give more examples, a formal and
abstract definition of adjoint functors is in order.
Definition: Let F :
C → D and G :
D → C be functors going in
opposite directions. F is a left adjoint to
G (G a right adjoint to F), denoted
by F
G, if
there exists natural transformations η :
I_{C} → GF and
ξ : FG → I_{D}, such
that the composites
G 
η
G → 
GFG 
G
ξ → 
G 
and
F 
F
η → 
FGF 
ξ
F → 
F 
are the identity natural transformations. (For different but
equivalent definitions, see Mac Lane 1971 or 1998, chap. IV.)
Here are some of the important facts regarding adjoint functors.
Firstly, adjoints are unique up to isomorphism; that is any two left
adjoints F and F' of a functor G are
naturally isomorphic. Secondly, the notion of adjointness is formally
equivalent to the notion of a universal morphism (or construction) and
to that of representable functor. (See, for instance Mac Lane
1998, chap. IV.) Each and every one of these notions exhibit an
aspect of a given situation. Thirdly, a left adjoint preserves all the
colimits which exist in its domain, and, dually, a right adjoint
preserves all the limits which exist in its domain.
We now give some examples of adjoint situations to illustrate the
pervasiveness of the notion.
 Instead of having a forgetful functor going into the category of
sets, in some cases only a part of the structure is forgotten. Here are
two standard examples:
 There is an obvious forgetful functor U :
AbGrp → AbMon from the category of abelian
groups to the category of abelian monoids: U forgets about
the inverse operation. The functor U has a left adjoint
F: AbMon → AbGrp which, given an
abelian monoid M, assigns to it the best possible abelian
group F(M) such that M can be embedded in
F(M) as a submonoid. For instance, if M
is
, then
F()
"is", that is, it
is isomorphic to
.
 Similarly, there is an obvious forgetful functor
U : Haus → Top from the category
of Hausdorff topological spaces to the category of topological spaces
which forgets the Hausdorff condition. Again, there is a functor
F : Top → Haus such that
F
U. Given a topological space X,
F(X) yields the best Hausdorff space constructed from
X: it is the quotient of X by the closure of the
diagonal
_{X}
⊆ X × X, which is an equivalence relation. In
contrast with the previous example where we had an embedding, this
time we get a quotient of the original structure.
 Consider now the category of compact Hausdorff spaces
kHaus and the forgetful functor U :
kHaus → Top, which forgets the compactness
property and the separation property. The left adjoint to this
U is the StoneCech compactification.
 There is a forgetful functor U :
Mod_{R} →
AbGrp from a category of Rmodules to the
category of abelian groups, where R is a commutative ring
with unit. The functor U forgets the action of R on
a group G. The functor U has both a left and a right
adjoint. The left adjoint is R
− : AbGrp →
Mod_{R} which sends an abelian group
G to the tensor product R
G and the right adjoint is given by the functor
Hom(R, −) : AbGrp
→ Mod_{R} which assigns to any group
G the modules of linear mappings
Hom(R, G).
 The case where the categories C and
D are posets deserves special attention here. Adjoint
functors in this context are usually called Galois
connections. Let C be a poset. Consider the
diagonal functor Δ : C → C ×
C, with Δ(X) = ‹X,
X› and for f :
X → Y, Δ(f) =
‹f,
f› : ‹X,
X› → ‹Y, Y›. In
this case, the leftadjoint to Δ is the coproduct, or the
sup, and the rightadjoint to Δ is the product, or the inf.
The adjoint situation can be described in the following special form:
X
Y ≤ Z
X ≤ Z, Y ≤ Z 


Z ≤ X
Y
Z ≤ Y, Z ≤ X 

where the vertical double arrow can be interpreted as rules of
inference going in both directions.
 Implication can also be introduced. Consider a functor with a
parameter: (−
X) : C → C. It
can easily be verified that when C is a poset, the
function (−
X) is order preserving and therefore a functor. A right adjoint to
(−
X) is a functor that
yields the largest element of C such that its infimum
with X is smaller than Z. This element is sometimes
called the relative pseudocomplement of X or, more commonly,
the implication. It is denoted by X ⇒ Z or by
X ⊃ Z. The adjunction can be presented as follows:
Y
X ≤ Z
Y ≤ X ⇒ Z 

 The negation operator ¬X can be introduced from the
last adjunction. Indeed, let Z be the bottom element ⊥ of
the lattice. Then, since Y
X ≤ ⊥ is always true, it follows that
Y ≤ X ⇒ ⊥ is also always true. But since X
⇒ ⊥ ≤ X is always the case, we get at the numerator
that X ⇒ ⊥
X = ⊥. Hence, X ⇒ ⊥ is
the largest element disjoint from X. We can therefore put
¬X =_{def} X ⇒ ⊥.
 Limits, colimits, and all the fundamental constructions of
category theory can be described as adjoints. Thus, products and
coproducts are adjoints, as are equalizers, coequalizers, pullbacks
and pushouts, etc. This is one of the reasons adjointness is central
to category theory itself: because all the fundamental operations of
category theory arise from adjoint situations.
 An equivalence of categories is a special case of
adjointness. Indeed, if in the above triangular identities the arrows
η : I_{C} →
GF and ξ : FG →
I_{D} are natural
isomorphisms, then the functors F and G
constitute an equivalence of categories. In practice, it is
the notion of equivalence of categories that matters and not the
notion of isomorphism of categories.
It is easy to prove certain facts about these operations directly from
the adjunctions. Consider, for instance, implication. Let Z =
X. Then we get at the numerator that Y
X ≤ X,
which is always true in a poset (as is easily verified). Hence,
Y ≤ X ⇒ X is also true for all Y and this is
only possible if X ⇒ X =
,
the top element of the lattice. Not only can logical operations be
described as adjoints, but they naturally arise as adjoints to basic
operations. In fact, adjoints can be used to define various
structures, distributive lattices, Heyting algebras, Boolean algebras,
etc. (See Wood, 2004.) It should be clear from the simple foregoing
example how the formalism of adjointness can be used to give syntactic
presentations of various logical theories. Furthermore, and this is a
key element, the standard universal and existential quantifiers can be
shown to be arising as adjoints to the operation of
substitution. Thus, quantifiers are on a par with the other logical
operations, in sharp contrast with the other algebraic approaches to
logic. (See, for instance Awodey 1996 or Mac Lane &
Moerdijk 1992.) More generally, Lawvere showed how syntax and
semantics are related by adjoint functors. (See Lawvere 1969b.)
Dualities play an important role in mathematics and they can be
described with the help of equivalences between categories. In other
words, many important mathematical theorems can be translated as
statements about the existence of adjoint functors, sometimes
satisfying additional properties. This is sometimes taken as
expressing the conceptual content of the theorem. Consider
the following fundamental case: let C be the category
whose objects are the locally compact abelian groups and the morphisms
are the continuous group homomorphisms. Then, the Pontryagin duality
theorem amounts to the claim that the category C is
equivalent to the category C°, that is, to the
opposite category. Of course, the precise statement requires that we
describe the functors F : C →
C° and G : C° →
C and prove that they constitute an equivalence of
categories.
Another well known and important duality was discovered by Stone in
the thirties and now bears his name. In one direction, an arbitrary
Boolean algebra yields a topological space, and in the other
direction, from a (compact Hausdorff and totally disconnected)
topological space, one obtains a Boolean algebra. Moreover, this
correspondence is functorial: any Boolean homomorphism is sent to a
continuous map of topological spaces, and, conversely, any continuous
map between the spaces is sent to a Boolean homomorphism. In other
words, there is an equivalence of categories between the category of
Boolean algebras and the dual of the category of Boolean spaces (also
called Stone spaces). (See Johnstone 1982 for an excellent
introduction and more developments.) The connection between a category
of algebraic structures and the opposite of a category of topological
structures established by Stone's theorem constitutes but one example
of a general phenomenon that did attract and still attracts a great
deal of attention from category theorists. Categorical study of
duality theorems is still a very active and significant field, and is
largely inspired by Stone's result. (For recent applications in logic,
see, for instance Makkai 1987, Taylor 2000, 2002a, 2002b.)
It is difficult to do justice to the short but intricate history of the
field. In particular it is not possible to mention all those who have
contributed to its rapid development. With this word of caution out of
the way, we will look at some of the main historical threads.
Categories, functors, natural transformations, limits and colimits
appeared almost out of nowhere in a paper by Eilenberg &
Mac Lane (1945) entitled "General Theory of Natural
Equivalences." We say "almost," because their earlier paper (1942)
contains specific functors and natural transformations at work,
limited to groups. A desire to clarify and abstract their 1942 results
led Eilenberg & Mac Lane to devise category theory. The
central notion at the time, as their title indicates, was that of
natural transformation. In order to give a general definition of the
latter, they defined functor, borrowing the term from Carnap, and in
order to define functor, they borrowed the word ‘category’
from the philosophy of Aristotle, Kant, and C. S. Peirce, but
redefining it mathematically.
After their 1945 paper, it was not clear that the concepts of category
theory would amount to more than a convenient language; this indeed
was the status quo for about fifteen years. Category theory was
employed in this manner by Eilenberg & Steenrod (1952), in an
influential book on the foundations of algebraic topology, and by
Cartan & Eilenberg (1956), in a ground breaking book on
homological algebra. (Curiously, although Eilenberg & Steenrod
defined categories, Cartan & Eilenberg simply assumed them!) These
books allowed new generations of mathematicians to learn algebraic
topology and homological algebra directly in the categorical language,
and to master the method of diagrams. Indeed, without the method of
diagram chasing, many results in these two books seem inconceivable,
or at the very least would have required a considerably more intricate
presentation.
The situation changed radically with Grothendieck's (1957) landmark
paper entitled "Sur quelques points d'algèbre homologique", in
which the author employed categories intrinsically to define and
construct more general theories which he (Grothendieck 1957) then
applied to specific fields, e.g., to algebraic geometry. Kan (1958)
showed that adjoint functors subsume the important concepts of limits
and colimits and could capture fundamental concepts in other areas (in
his case, homotopy theory).
At this point, category theory became more than a convenient
language, by virtue of two developments.
 Employing the axiomatic method and the language of categories,
Grothendieck (1957) defined in an abstract fashion types of
categories, e.g., additive and Abelian categories, showed how to
perform various constructions in these categories, and proved various
results about them. In a nutshell, Grothendieck showed how to develop
part of homological algebra in an abstract setting of this sort. From
then on, a specific category of structures, e.g., a category of
sheaves over a topological space X, could be seen as a token
of an abstract category of a certain type, e.g., an Abelian
category. One could therefore immediately see how the methods of,
e.g., homological algebra could be applied to, for instance, algebraic
geometry. Furthermore, it made sense to look for other types of
abstract categories, ones that would encapsulate the fundamental and
formal aspects of various mathematical fields in the same way that
Abelian categories encapsulated fundamental aspects of homological
algebra.
 Thanks in large part to the efforts of Freyd and Lawvere, category
theorists gradually came to see the pervasiveness of the concept of
adjoint functors is. Not only does the existence of adjoints to given
functors permit definitions of abstract categories (and presumably
those which are defined by such means have a privileged status) but as
we mentioned earlier, many important theorems and even theories in
various fields can be seen as equivalent to the existence of specific
functors between particular categories. By the early 1970's, the
concept of adjoint functors was seen as central to category
theory.
With these developments, category theory became an autonomous field of
research, and pure category theory could be developed. And indeed, it
did grow rapidly as a discipline, but also in its applications, mainly
in its source contexts, namely algebraic topology and homological
algebra, but also in algebraic geometry and, after the appearance of
Lawvere's Ph. D thesis, in universal algebra. This thesis also
constitutes a landmark in this history of the field, for in it Lawvere
proposed the category of categories as a foundation for category
theory, set theory and, thus, the whole of mathematics, as well as
using categories for the study of the logical aspects of
mathematics.
Over the course of the 1960's, Lawvere outlined the basic framework
for an entirely original approach to logic and the foundations of
mathematics. He achieved the following:
 Axiomatized the category of sets (Lawvere 1964) and of categories
(Lawvere 1966);
 Gave a categorical description of theories that was independent of
syntactical choices and sketched how completeness theorems for logical
systems could be obtained by categorical methods (Lawvere 1967);
 Characterized Cartesian closed categories and showed their
connections to logical systems and various logical paradoxes (Lawvere
1969);
 Showed that the quantifiers and the comprehension schemes could be
captured as adjoint functors to given elementary operations (Lawvere
1966, 1969, 1970, 1971);
 Argued that adjoint functors should generally play a major
foundational role through the notion of "categorical doctrines"
(Lawvere 1969).
Meanwhile, Lambek (1968, 1969, 1972) described categories in terms of
deductive systems and employed categorical methods for
prooftheoretical purposes.
All this work culminated in another notion, thanks to Grothendieck and
his school: that of a topos. Even though toposes appeared in
the 1960's, in the context of algebraic geometry, again from the mind
of Grothendieck, it was certainly Lawvere and Tierney's (1972)
elementary axiomatization of a topos which gave impetus to its
attaining foundational status. Very roughly, a topos is a category
possessing a logical structure sufficiently rich to develop most of
"ordinary mathematics", that is, most of what is taught to mathematics
undergraduates. As such, a topos can be thought of as a categorical
theory of sets. But it is also a generalized topological space, thus
providing a direct connection between logic and geometry.
The 1970s saw the development and application of the topos concept in
many different directions. The very first applications outside
algebraic geometry were in set theory, where various independence
results were recast in terms of topos (Tierney 1972, Bunge 1974, but
also Blass & Scedrov 1989, Blass & Scedrov 1992, Freyd 1980,
Mac Lane & Moerdijk 1992, Scedrov 1984). Connections with
intuitionistic mathematics were noted early on, and toposes are still
used to investigate models of various aspects of intuitionism (Lambek
& Scott 1986, Mac Lane & Moerdijk 1992, Van der Hoeven
& Moerdijk 1984a, 1984b, 1984c, Moerdijk 1984, Moerdijk 1995a,
Moerdijk 1998, Moerdijk & Palmgren 1997, Moerdijk & Palmgren
2002). For more on the history of topos theory, see Mc Larty
(1992).
More recently, topos theory has been employed to investigate various
forms of constructive mathematics or set theory (Joyal & Moerdijk
1995, Taylor 1996), recursiveness, and models of higherorder type
theories generally. The introduction of the socalled "effective
topos" and the search for axioms for synthetic domain theory are worth
mentioning (Hyland 1982, Hyland 1988, 1991, Hyland et
al. 1990, Mc Larty 1992, Jacobs 1999, Van Oosten 2002 and
the references therein). Lawvere's early motivation was to provide a
new foundation for differential geometry, a lively research area which
is now called "synthetic differential geometry" (Lawvere 2000, 2002,
Kock 1981, Bell 1988, 1995, 1998, Moerdijk & Reyes 1991). This is
only the tip of the iceberg; toposes could prove to be for the 21st
century what Lie groups were to the 20th century.
From the 1980s to the present, category theory has found new
applications. In theoretical computer science, category theory is now
firmly rooted, and contributes, among other things, to the development
of new logical systems and to the semantics of programming. (Pitts
2000, Plotkin 2000, Scott 2000, and the references therein). Its
applications to mathematics are becoming more diverse, even touching
on theoretical physics, which employs higherdimensional category
theory — which is to category theory what higherdimensional
geometry is to plane geometry — to study the socalled "quantum
groups" and quantum field theory (Baez & Dolan 2001 and other
publications by these authors).
Category theory challenges philosophers in two ways, which are not
necessarily mutually exclusive. On the one hand, it is certainly the
task of philosophy to clarify the general epistemological and
ontological status of categories and categorical methods, both in the
practice of mathematics and in the foundational landscape. On the
other hand, philosophers and philosophical logicians can employ
category theory and categorical logic to explore philosophical and
logical problems. I now discuss these challenges, briefly, in turn.
Category theory is now a common tool in the mathematician's toolbox;
that much is clear. It is also clear that category theory organizes
and unifies much of mathematics. (See for instance Mac Lane
1971, 1998 or Pedicchio & Tholen 2004.) No one will deny these
simple facts.
Doing mathematics in a categorical framework is almost always
radically different from doing it in a settheoretical framework (the
exception being working with the internal language of a Boolean topos;
whenever the topos is not Boolean, then the main difference lies in
the fact that the logic is intuitionistic). Hence, as is
often the case when a different conceptual framework is adopted, many
basic issues regarding the nature of the objects studied, the nature
of the knowledge involved, and the nature of the methods used have to
be reevaluated. We will take up these three aspects in turn.
Two facets of the nature of mathematical objects within a categorical
framework have to be emphasized. First, objects are always given in a
category. An object exists in and depends upon an ambient
category. Furthermore, an object is characterized by the morphisms
going in it and/or the morphisms coming out of it. Second, objects are
always characterized up to isomorphism (in the best cases, up to a
unique isomorphism). There is no such thing, for instance, as
the natural numbers. However, it can be argued that there is
such a thing as the concept of natural numbers. Indeed, the
concept of natural numbers can be given unambiguously, via the
DedekindPeanoLawvere axioms, but what this concept refers to in
specific cases depends on the context in which it is interpreted,
e.g., the category of sets or a topos of sheaves over a topological
space. It is hard to resist the temptation to think that category
theory embodies a form of structuralism, that it describes
mathematical objects as structures since the latter, presumably, are
always characterized up to isomorphism. Thus, the key here has to do
with the kind of criterion of identity at work within a categorical
framework and how it resembles any criterion given for objects which
are thought of as forms in general. One of the standard objections
presented against this view is that if objects are thought of as
structures and only as abstract structures, meaning here that
they are separated from any specific or concrete representation, then
it is impossible to locate them within the mathematical universe. (See
Hellman 2003 for a standard formulation of the objection,
Mc Larty 1993, Awodey 2004, Landry & Marquis 2005 for
relevant material on the issue.) A slightly different way to make
sense of the situation is to think of mathematical objects as
types for which there are tokens given in different
contexts. This is strikingly different from the situation one finds in
set theory, in which mathematical objects are defined uniquely and
their reference is given directly. Although one can make room for
types within set theory via equivalence classes or isomorphism types
in general, the basic criterion of identity within that
framework is given by the axiom of extensionality and thus,
ultimately, reference is made to specific sets. Furthermore, it can
be argued that the relation between a type and its token is
not represented adequately by the membership relation. A
token does not belong to a type, it is not an element of a type, but
rather it is an instance of it. In a categorical framework, one always
refers to a token of a type, and what the theory
characterizes directly is the type, not the tokens. In this framework,
one does not have to locate a type, but tokens of it are, at least in
mathematics, epistemologically required. This is simply the reflection
of the interaction between the abstract and the concrete in the
epistemological sense (and not the ontological sense of these latter
expressions.) (See Ellerman 1988, Marquis 2000 and Marquis
forthcoming.)
The history of category theory offers a rich source of information to
explore and take into account for an historically sensitive
epistemology of mathematics. It is hard to imagine, for instance, how
algebraic geometry and algebraic topology could have become what they
are now without categorical tools. (See, for instance, Corfield 2003,
Mc Larty 1994.) Category theory has lead to
reconceptualizations of various areas of mathematics based on purely
abstract foundations. Moreover, when developed in a categorical
framework, traditional boundaries between disciplines are shattered
and reconfigured; to mention but one important example, topos theory
provides a direct bridge between algebraic geometry and logic, to the
point where certain results in algebraic geometry are directly
translated into logic and vice versa. Certain concepts that were
geometrical in origin are more clearly seen as logical (for example,
the notion of coherent topos). Algebraic topology also lurks in the
background. On a different but important front, it can be argued that
the distinction between mathematics and metamathematics cannot be
articulated in the way it has been. All these issues have to be
reconsidered and reevaluated.
Moving closer to mathematical practice, category theory allowed for
the development of methods that have changed and continue to change
the face of mathematics. It could be argued that category theory
represents the culmination of one of deepest and most powerful
tendencies in twentieth century mathematical thought: the search for
the most general and abstract ingredients in a given
situation. Category theory is, in this sense, the legitimate heir of
the DedekindHilbertNoetherBourbaki tradition, with its emphasis on
the axiomatic method and algebraic structures. When used to
characterize a specific mathematical domain, category theory reveals
the frame upon which that area is built, the overall structure
presiding to its stability, strength and coherence. The structure of
this specific area, in a sense, might not need to rest on anything,
that is, on some solid soil, for it might very well be just one part
of a larger network that is without any Archimedean point, as if
floating in space. To use a wellknown metaphor: from a categorical
point of view, Neurath's ship has become a spaceship.
Still, it remains to be seen whether category theory should be "on the
same plane," so to speak, as set theory, whether it should be taken as
a serious alternative to set theory as a foundation for mathematics,
or whether it is foundational in a different sense altogether. (That
this very question applies even more forcefully to topos theory will
not detain us.)
Lawvere from early on promoted the idea that a category of categories
could be used as a foundational framework. (See Lawvere 1964, 1966.)
This proposal now rests in part on the development of
higherdimensional categories, also called weak ncategories.
(See, for instance Makkai 1998.) The advent of topos theory in the
seventies brought new possibilities. Mac Lane has suggested
that certain toposes be considered as a genuine foundation for
mathematics. (See Mac Lane 1986.) Lambek proposed the socalled
free topos as the best possible framework, in the sense that
mathematicians with different philosophical outlooks might nonetheless
agree to adopt it. (See Couture & Lambek 1991, 1992, Lambek
1994.) He has recently argued that there is no topos that can
thoroughly satisfy a classical mathematician. (See Lambek 2004.) (For
more on the various foundational views among category theorists, see
Landry & Marquis 2005.)
Arguments have been advanced for and against category theory as a
foundational framework. (Blass 1984 surveys the relationships between
category theory and set theory. Feferman 1977, Bell 1981, and Hellman
2003 argue against category theory. See Marquis 1995 for a quick
overview and proposal and Mc Larty 2004 and Awodey 2004 for
replies to Hellman 2003.) This matter is further complicated by the
fact that the foundations of category theory itself have yet to be
clarified. For there may be many different ways to think of a universe
of higherdimensional categories as a foundations for mathematics. An
adequate language for such a universe still has to be presented
together with definite axioms for mathematics. (See Makkai 1998 for a
short description of such a language.)
It is an established fact that category theory is employed to study
logic and philosophy. Indeed, categorical logic, the study of logic by
categorical means, has been under way for about 30 years now and still
vigorous. Some of the philosophically relevant results obtained in
categorical logic are:
 The hierarchy of categorical doctrines: regular categories,
coherent catgories, Heyting categories and Boolean categories; all
these correspond to welldefined logical systems, together with
deductive systems and completeness theorems; they suggest that logical
notions, including quantifiers, arise naturally in a specific order
and are not haphazardly organized;
 Joyal's generalization of KripkeBeth semantics for intuitionistic
logic to sheaf semantics (Lambek & Scott 1986, Mac Lane
& Moerdijk 1992);
 Coherent and geometric logic, socalled, whose practical and
conceptual significance has yet to be explored (Makkai & Reyes
1977, Mac Lane & Moerdiejk 1992, Johnstone 2002);
 The notions of generic model and classifying topos of a theory
(Makkai & Reyes 1977, Boileau & Joyal 1981, Bell 1988, Mac Lane
& Moerdijk 1992, Johnstone 2002);
 The notion of strong conceptual completeness and the associated
theorems (Makkai & Reyes 1977, Butz & Moerdijk 1999, Makkai
1981, Pitts 1989, Johnstone 2002);
 Geometric proofs of the independence of the continuum hypothesis
and other strong axioms of set theory (Tierney 1972, Bunge 1974, Freyd
1980, 1987, Blass & Scedrov 1983, 1989, 1992, Mac Lane &
Moerdijk 1992);
 Models and development of constructive mathematics (see
bibliography below);
 Synthetic differential geometry, an alternative to standard and
nonstandard analysis (Kock 1981, Bell 1998, 2001);
 The construction of the socalled effective topos, in which every
function on the natural numbers is recursive (Mc Larty 1992,
Hyland 1982, 1991, Van Oosten 2002);
 Categorical models of linear logic, modal logic, fuzzy sets, and
general higherorder type theories (Reyes 1991, Reyes & Zawadoski
1993, Reyes & Zolfaghari 1991, 1996, Makkai & Reyes 1995,
Ghilardi & Zawadowski 2002, Rodabaugh & Klement 2003, Jacobs
1999, Taylor 1999, Johnstone 2002, Blute & Scott 2004);
 A graphical syntax called "sketches" (Barr & Wells 1985, 1999,
Makkai 1997a, 1997b, 1997c, Johnstone 2002).
Categorical tools in logic offer considerable flexibility, as is
illustrated by the fact that almost all the surprising results of
constructive and intuitionistic mathematics can be modeled in a proper
categorical setting. At the same time, the standard settheoretic
notions, e.g. Tarski's semantics, have found natural generalizations
in categories. Thus, categorical logic has roots in logic as it was
developed in the twentieth century, while at the same time providing a
powerful and novel framework with numerous links to other parts of
mathematics.
Category theory also bears on more general philosophical questions.
From the foregoing disussion, it should be obvious that category
theory and categorical logic ought to have an impact on almost all
issues arising in philosophy of logic: from the nature of identity
criteria to the question of alternative logics, category theory always
sheds a new light on these topics. Similar remarks can be made when we
turn to ontology, in particular formal ontology: the part/whole
relation, boundaries of systems, ideas of space, etc. Ellerman (1988)
has bravely attempted to show that category theory constitutes a
theory of universals, one having properties radically different from
set theory, which is also seen as a theory of universals. Moving from
ontology to cognitive science, MacNamara & Reyes (1994) have tried
to employ categorical logic to provide a different logic of
reference. In particular, they have attempted to clarify the
relationships between count nouns and mass terms. Other researchers
are using category theory to study complex systems, cognitive neural
networks, and analogies. (See, for instance, Ehresmann &
Vanbremeersch 1987, Healy 2000, ArziGonczarowski 1999.)
Category theory offers thus many philosophical challenges, challenges
which will hopefully be taken up in years to come.
The following Bibliography has been divided into sections to
facilitate programmatic reading. However, to more easily lookup a
citation in the text, the reader may wish to use the following
document, which has been compiled and alphabetically sorted:
Alphabetically Sorted, Complete Bibliography
A. General books on category theory
A.1 Introductory texts
 Lawvere, F. W. & Rosebrugh, R., 2003, Sets for
Mathematics, Cambridge: Cambridge University Press.
 Lawvere, F. W. & Schanuel, S., 1997, Conceptual
Mathematics: A First Introduction to Categories, Cambridge:
Cambridge University Press.
These two texts are intended for undergraduates or nonspecialists.
A.2 Intermediate texts
 Birkoff, G. & Mac Lane, S., 1999, Algebra, 3^{rd} ed.,
Providence: AMS.
Much more than category theory in this classic textbook, but it
contains the rudiments of the theory with applications.
The following book provides an accessible approach to category
theory and categorical logic:
 Mc Larty, C., 1992, Elementary Categories, Elementary
Toposes, Oxford: Oxford University Press.
The standard reference on category theory is unquestionably:
 Mac Lane, S., 1998, Categories for the Working
Mathematician, 2^{nd} edition, New York:
SpringerVerlag.
A.3 Other interesting and useful books
 Adamek, J. et al., 1990, Abstract and Concrete
Categories: The Joy of Cats, New York: Wiley.
 Borceux, F., 1994, Handbook of Categorical Algebra, 3
volumes, Cambridge: Cambridge University Press.
 Freyd, P., 1990, Categories, Allegories, Amsterdam: North
Holland.
 Hatcher, W. S., 1982, The Logical Foundations of
Mathematics, Oxford: Pergamon Press.
 Pareigis, B., 1970, Categories and Functors, New York:
Academic Press.
 Pedicchio, M. C. & Tholen, W., 2004, Categorical
Foundations, Cambridge: Cambridge University Press.
B. Books and articles on higherdimensional category theory
 Baez, J., 1997, "An Introduction to nCategories",
Category Theory and Computer Science, Lecture Notes in
Computer Science, 1290, Berlin: SpringerVerlag,
1–33.
 Baez, J. & Dolan, J., 1998a, "HigherDimensional Algebra III.
nCategories and the Algebra of Opetopes", Advances in
Mathematics, 135, 145–206.
 Baez, J. & Dolan, J., 1998b, "Categorification", Higher
Category Theory, Contemporary Mathematics, 230,
Providence: AMS, 1–36.
 Baez, J. & Dolan, J., 2001, "From Finite Sets to Feynman
Diagrams", Mathematics Unlimited – 2001 and Beyond,
Berlin: Springer, 29–50.
 Batanin, M., 1998, "Monoidal Globular Categories as a Natural
Environment for the Theory of Weak nCategories", Advances
in Mathematics, 136, 39–103.
 Hermida, C. & Makkai, M. & Power, J., 2000, "On Weak
Higherdimensional Categories I", Journal of Pure and Applied
Algebra, 154, no. 13, 221–246.
 Hermida, C. & Makkai, M. & Power, J., 2001, "On Weak
Higherdimensional Categories 2", Journal of Pure and Applied
Algebra, 157, no. 23, 247–277.
 Hermida, C. & Makkai, M. & Power, J., 2002, "On Weak
Higherdimensional Categories 3", Journal of Pure and Applied
Algebra, 166, no. 12, 83–104.
 Leinster, T., 2002, "A Survey of Definitions of
ncategories", Theory and Applications of Categories,
(electronic), 10, 1–70.
C. Articles with a philosophical bent
C.1 Category theory and the philosophy of mathematics
 Awodey, S., 1996, "Structure in Mathematics and Logic: A
Categorical Perspective", Philosophia Mathematica,
3, 209–237.
 Awodey, S., 2004, "An Answer to Hellman's Question: Does Category
Theory Provide a Framework for Mathematical Structuralism",
Philosophia Mathematica, 12,
54–64.
 Biss, D.K., 2003, "Which Functor is the Projective Line?",
American Mathematical Monthly, 110, 7,
574–592.
 Couture, J. & Lambek, J., 1991, "Philosophical Reflections on
the Foundations of Mathematics", Erkenntnis,
34, 2, 187–209.
 Couture, J. & Lambek, J., 1992, "Erratum:"Philosophical
Reflections on the Foundations of Mathematics"", Erkenntnis,
36, 1, 134.
 Hellman, G., 2003, "Does Category Theory Provide a Framework for
Mathematical Structuralism?", Philosophia Mathematica,
11, 2, 129–157.
 Lambek, J., 1982, "The Influence of Heraclitus on Modern
Mathematics", Scientific Philosophy Today, J. Agassi and R.S.
Cohen, eds., Dordrecht, Reidel, 111–122.
 Lambek, J. 1994, "Are the Traditional Philosophies of Mathematics
Really Incompatible?", Mathematical Intelligencer,
16, 1, 56–62.
 Landry, E., 1999, "Category Theory: the Language of Mathematics",
Philosophy of Science, 66, 3: supplement,
S14–S27.
 Landry, E., 2001, "Logicism, Structuralism and Objectivity",
Topoi, 20, 1, 79–95.
 Landry, E. & Marquis, J.P., 2005, "Categories in Context:
Historical, Foundational and philosophical", Philosophia
Mathematica, 13, 1–43.
 Mac Lane, S., 1981, "Mathematical Models: a Sketch for the
Philosophy of Mathematics", American Mathematical Monthly,
88, 7, 462–472.
 Mac Lane, S., 1986, Mathematics, Form and Function,
New York: Springer.
 Mac Lane, S., 1996, "Structure in Mathematics. Mathematical
Structuralism", Philosophia Mathematica, 4,
2, 174–183.
 Makkai, M., 1999, "On Structuralism in Mathematics", Language,
Logic and Concepts, Cambridge: MIT Press, 43–66.
 Marquis, J.P., 1993, "Russell's Logicism and Categorical
Logicisms", Russell and Analytic Philosophy, A. D. Irvine
& G. A. Wedekind, (eds.), Toronto, University of Toronto Press,
293–324.
 Marquis, J.P., 2000, "Three Kinds of Universals in Mathematics?",
in Logical Consequence: Rival Approaches and New Studies in Exact
Philosophy: Logic, Mathematics and Science, Vol. II, B. Brown
& J. Woods, eds., Oxford: Hermes, 191212, 2000
 Marquis, J.P., forthcoming, "Categories, Sets and the Nature of
Mathematical Entities", in The Age of Alternative Logics. Assessing
philosophy of logic and mathematics today, J. van Benthem, G.
Heinzmann, Ph. Nabonnand, M. Rebuschi, H.Visser, eds., Kluwer.
 Mc Larty, C., 1993, "Numbers Can be Just What They Have to",
Noûs, 27, 487–498.
 Mc Larty, C., 1994, "Category Theory in Real Time",
Philosophia Mathematica, 2, no. 1,
36–44.
 Mc Larty, C., 2004, "Exploring Categorical Structuralism",
Philosophia Mathematica, 12,
37–53.
 Mc Larty, C., 2005, "Learning from Questions on Categorical
Foundations", Philosophia Mathematica, 13, 1,
44–60.
 Shapiro, S., 2005, "Categories, Structures and the FregeHilbert
Controversy: the Status of Metamathematics", Philosophia
Mathematica, 13, 1, 61–77.
C.2 Category theory and set theory
 Bell, J. L., 1981, "Category Theory and the Foundations of
Mathematics", British Journal for the Philosophy of Science,
32, 349–358.
 Bell, J. L., 1982, "Categories, Toposes and Sets",
Synthese, 51, 3, 293–337.
 Bell, J. L., 1986, "From Absolute to Local Mathematics",
Synthese, 69, 3, 409–426.
 Blass, A., 1984, "The Interaction Between Category Theory and Set
Theory", Mathematical Applications of Category Theory,
30, Providence: AMS, 5–29.
 Feferman, S., 1977, "Categorical Foundations and Foundations of
Category Theory", Logic, Foundations of Mathematics and
Computability, R. Butts (ed.), Reidel, 149–169.
 Mac Lane, S., 1969, "Foundations for Categories and Sets",
Category Theory, Homology Theory and their Applications II,
Berlin: Springer, 146–164.
 Mac Lane, S., 1969, "One Universe as a Foundation for
Category Theory", Reports of the Midwest Category Seminar III,
Berlin: Springer, 192–200.
 Mac Lane, S., 1971, "Categorical algebra and SetTheoretic
Foundations", Axiomatic Set Theory, Providence: AMS,
231–240.
 Marquis, J.P., 1995, "Category Theory and the Foundations of
Mathematics: Philosophical Excavations", Synthese,
103, 421–447.
 Moerdijk, I., 1998, "Sets, Topoi and Intuitionism", Philosophia
Mathematica, 6, no. 2, 169–177.
C.3 Categorical foundations of logic and mathematics
 Bell, J. L., 1988, "Infinitesimals", Synthese,
75, 3, 285–315.
 Bell, J. L., 1995, "Infinitesimals and the Continuum",
Mathematical Intelligencer, 17, 2,
55–57.
 Bell, J. L., 2001, "The Continuum in Smooth Infinitesimal
Analysis", Reuniting the Antipodes — Constructive and
Nonstandard Views on the Continuum, Synthese Library,
306, Dordrecht: Kluwer, 19–24.
 Johnstone, P. T., 1985, "How General is a Generalized
Space?", Aspects of Topology, Cambridge: Cambridge University
Press, 77–111.
 Lambek, J., 1994, "What is a Deductive System?", What is a
Logical System?, Studies in Logic and Computation,
4, Oxford: Oxford University Press,
141–159.
 Lawvere, F. W., 2003, "Foundations and Applications:
Axiomatization and Education. New Programs and Open Problems in the
Foundation of Mathematics", Bullentin of Symbolic Logic,
9, 2, 213–224.
 Mac Lane, S., 1997, "Categorical Foundations of the Protean
Character of Mathematics", Philosophy of Mathematics Today,
Dordrecht: Kluwer, 117–122.
 Makkai, M., 1998, "Towards a Categorical Foundation of
Mathematics", Lecture Notes in Logic, 11,
Berlin: Springer, 153–190.
 Mc Larty, C., 1991, "Axiomatizing a Category of Categories",
Journal of Symbolic Logic, 56, no. 4,
1243–1260.
 Wood, R.J., 2004, " Ordered Sets via Adjunctions", Categorical
Foundations, M. C. Pedicchio & W. Tholen, eds., Cambridge:
Cambridge University Press.
C.4 Applications of category theory to philosophical of cognitive
issues
 Ellerman, D., 1988, "Category Theory and Concrete Universals",
Synthese, 28, 409–429.
 La Palme Reyes, M., et. al., 1994, "The nonBoolean Logic of
Natural Language Negation", Philosophia Mathematica,
2, no. 1, 45–68.
 La Palme Reyes, M., et. al., 1999, "Count Nouns, Mass Nouns, and
their Transformations: a Unified Categorytheoretic Semantics",
Language, Logic and Concepts, Cambridge: MIT Press,
427–452.
 Lambek, J., 2004, "What is the world of Mathematics? Provinces of
Logic Determined", Annals of Pure and Applied Logic,
126(13), 149–158.
 Lawvere, F. W., 1992, "Categories of Space and of Quantity",
The Space of Mathematics, Foundations of Communication and
Cognition, Berlin: De Gruyter, 14–30.
 Lawvere, F. W., 1994, "Cohesive Toposes and Cantor's lauter
Ensein ", Philosophia Mathematica, 2, 1,
5–15.
 Lawvere, F. W., 1994, "Tools for the Advancement of
Objective Logic: Closed Categories and Toposes", The Logical
Foundations of Cognition, Vancouver Studies in Cognitive Science,
4, Oxford: Oxford University Press, 43–56.
 MacNamara, J. & Reyes, G., (eds.), 1994, The Logical
Foundation of Cognition, Oxford: Oxford University Press.
 Marquis, J.P., 2000, "Three Kinds of Universals in Mathematics?",
Logical Consequence: Rival Approaches and New Studies in Exact
Philosophy: Logic, Mathematics and Science, Vol. 2, Oxford:
Hermes, 191–212.
 Pitts, A. M. & Taylor, P., 1989, "A Note of Russell's
Paradox in Locally Cartesian Closed Categories", Studia
Logica, 48, no. 3, 377–387.
D. Books and papers on categorical logic
D.1 Introductoy books and papers
One of the first introductory books on the subject is and still
extremely useful for readers without a strong background in category
theory:
 Goldblatt, R., 1979, Topoi: The Categorical Analysis of Logic,
Studies in logic and the foundations of mathematics, Amsterdam:
Elsevier.
For readers with a background in logic and set theory:
 Bell, J. L., 1988, Toposes and Local Set Theories: An
Introduction, Oxford: Oxford University Press.
 Bunge, M., 1984, "Toposes in Logic and Logic in Toposes",
Topoi, 3, no. 1, 13–22.
 Mac Lane, S., 1975, "Sets, Topoi, and Internal Logic in
Categories", Studies in Logic and the Foundations of
Mathematics, 80, Amsterdam: North Holland,
119–134.
D.2 Advanced textbooks
 Barr, M. & Wells, C., 1985, Toposes, Triples and
Theories, New York: SpringerVerlag.
 Jacobs, B., 1999, Categorical Logic and Type Theory,
Amsterdam: North Holland.
 Johnstone, P. T., 1977, Topos Theory, New York:
Academic Press.
 Johnstone, P. T., 2002a, Sketches of an Elephant: a
Topos Theory Compendium. Vol. 1, Oxford Logic Guides,
43, Oxford: Oxford University Press.
 Lambek, J. & Scott, P.J., 1986, Introduction to Higher
Order Categorical Logic, Cambridge: Cambridge University
Press.
 Mac Lane, S. & Moerdijk, I., 1992, Sheaves in
Geometry and Logic, New York: SpringerVerlag.
 Taylor, P., 1999, Practical Foundations of Mathematics,
Cambridge: Cambridge University Press.
Note that the second volume of (Johnstone 2002a) contains a clear
and exhaustive presentation of categorical logic in a topos theoretical
setting.
D.3 Research monographs
 Adamek, J. et al., 1994, Locally Presentable and
Accessible Categories, Cambridge: Cambridge University Press.
 Makkai, M. & Paré, R., 1989, Accessible Categories:
the Foundations of Categorical Model Theory, Contemporary
Mathematics 104, Providence: AMS.
 Makkai, M. & Reyes, G., 1977, FirstOrder Categorical
Logic, Springer Lecture Notes in Mathematics 611, New York:
Springer.
D.4 Books and introductory papers written for computer
scientists
 Barr, M. & Wells, C., 1999, Category Theory for Computing
Science, Montreal: CRM.
 Crole, R. L., 1994, Categories for Types, Cambridge:
Cambridge University Press.
 Peirce, B., 1991, Basic Category Theory for Computer
Scientists, Cambridge: MIT Press.
 Pitts, A. M., 2000, "Categorical Logic", Handbook of
Logic in Computer Science, Vol.5, Oxford: Oxford Unversity Press,
39–128.
 Plotkin, B., 2000, "Algebra, Categories and Databases",
Handbook of Algebra, Vol. 2, Amsterdam: Elsevier,
79–148.
 Scott, P.J., 2000, "Some Aspects of Categories in Computer
Science", Handbook of Algebra, Vol. 2, Amsterdam: North
Holland, 3–77.
D.5 A book on nonclassical logics from a categorical point of
view
 Ghilardi, S. & Zawadowski, M., 2002, Sheaves, Games &
Model Completions: A Categorical Approach to Nonclassical Porpositional
Logics, Dordrecht: Kluwer.
D.6 Research papers on various aspects of categorical logic
D.6.1 Completeness results
 Awodey, S. & Butz, C., 2000, "Topological Completeness for
Higher Order Logic", Journal of Symbolic Logic,
65, 3, 1168–1182.
 Boileau, A. & Joyal, A., 1981, "La logique des topos",
Journal of Symbolic Logic, 46, 1,
6–16.
 Galli, A. & Reyes, G. & Sagastume, M., 2000, "Completeness
Theorems via the Double Dual Functor", Studia Logical,
64, no. 1, 61–81.
 Makkai, M., 1987, "Stone Duality for FirstOrder Logic",
Advances in Mathematics, 65, 2,
97–170.
 Makkai, M., 1988, "Strong Conceptual Completeness for First Order
Logic", Annals of Pure and Applied Logic, 40,
167–215.
 Makkai, M., 1997a, "Generalized Sketches as a Framework for
Completeness Theorems I", Journal of Pure and Applied Algebra,
115, 1, 49–79.
 Makkai, M., 1997b, "Generalized Sketches as a Framework for
Completeness Theorems II", Journal of Pure and Applied
Algebra, 115, 2, 179–212.
 Makkai, M., 1997c, "Generalized Sketches as a Framework for
Completeness Theorems III", Journal of Pure and Applied
Algebra, 115, 3, 241–274.
 Makkei, M. & Reyes, G., 1995, "Completeness Results for
Intuitionistic and Modal Logic in a Categorical Setting", Annals of
Pure and Applied Logic, 72, 1, 25–101.
 Pitts, A. M., 1987, "Interpolation and Conceptual
Completeness for Pretoposes via Category Theory", Mathematical
Logic and Theoretical Computer Science, Lecture Notes in Pure and
Applied Mathematics, 106, New York: Dekker,
301–327.
 Pitts, A. M., 1989, "Conceptual Completeness for Firstorder
Intuitionistic Logic: an Application of Categorical Logic", Annals
of Pure and Applied Logic, 41, no. 1,
33–81.
D.6.2 Categorical logic and constructivism
 Freyd, P., Friedman, H. & Scedrov, A., 1987, "Lindembaum
Algebras of Intuitionistic Theories and Free Categories", Annals of
Pure and Applied Logic, 35, 2,
167–172.
 Johnstone, P. T., 1979a, "Conditions Related to De Morgan's
Law", Applications of Sheaves, Lecture Notes in Mathematics,
753, Berlin: Springer, 479–491.
 Johnstone, P. T., 1979b, "Another Condition Equivalent to De
Morgan's Law", Communications in Algebra, 7,
no. 12, 1309–1312.
 Johnstone, P. T., 1981, "Tychonoff's Theorem without the
Axiom of Choice", Fundamenta Mathematicae,
113, no. 1, 21–35.
 Johnstone, P. T., 1982, Stone Spaces,
Cambridge:Cambridge University Press.
 Lambek, J., 1989, "On the Sheaf of Possible Worlds",
Categorical Topology and its relation to Analysis, Algebra and
Combinatorics, Teaneck: World Scientific Publishing,
36–53.
 Lambek, J. & Scott, P.J., 1981, "Intuitionistic Type Theory and
Foundations", Journal of Philosophical Logic,
10, 1, 101–115.
 Lambek, J. & Scott, P.J., 1983, "New Proofs of Some
Intuitionistic Principles", Zeitschrift fÂ¸r
Mathematische Logik und Grundlagen der Mathematik,
29, 6, 493–504.
 Moerdijk, I., 1984, "HeineBorel does not imply the Fan Theorem",
Journal of Symbolic Logic, 49, no. 2,
514–519.
 Moerdijk, I., 1995a, "A Model for Intuitionistic Nonstandard
Arithmetic", Annals of Pure and Applied Logic,
73, no. 1, 37–51.
 Moerdijk, I. & Palmgren, E., 1997, "Minimal Models of Heyting
Arithmetic", Journal of Symbolic Logic, 62,
no. 4, 1448–1460.
 Moerdijk, I. & Palmgren, E., 2002, "Type Theories, Toposes and
Constructive Set Theory: Predicative Aspects of AST", Annals of
Pure and Applied Logic, 114, no. 1–3,
155–201.
 Pitts, A. M., 1992, "On an Interpretation of Secondorder
Quantification in Firstorder Propositional Intuitionistic Logic",
Journal of Symbolic Logic, 57, no. 1,
33–52.
 Seely, R. A. G., 1984, "Locally Cartesian Closed
Categories and Type Theory", Mathematical Proceedings of the
Cambridge Mathematical Society, 95, no. 1,
33–48.
 Taylor, P., 1996, "Intuitionistic sets and Ordinals", Journal
of Symbolic Logic, 61, 705–744.
 Van der Hoeven, G. & Moerdijk, I., 1984a, "Sheaf Models for
Choice Sequences", Annals of Pure and Applied Logic,
27, no. 1, 63–107.
 Van der Hoeven, G. & Moerdijk, I., 1984b, "On Choice Sequences
determined by Spreads", Journal of Symbolic Logic,
49, no. 3, 908–916.
 Van der Hoeven, G. & Moerdijk, I., 1984c, "Constructing Choice
Sequences for Lawless Sequences of Neighbourhood Functions", Models
and Sets, Lecture Notes in Mathematics, 1103,
Berlin: Springer, 207–234.
D.6.3 Applications of categorical logic to set theory
 Blass, A. & Scedrov, A., 1983, Classifying Topoi and Finite
Forcing , Journal of Pure and Applied Algebra,
28, 111–140.
 Blass, A. & Scedrov, A., 1989, Freyd's Model for the
Independence of the Axiom of Choice, Providence: AMS.
 Blass, A. & Scedrov, A., 1992, "Complete Topoi Representing
Models of Set Theory", Annals of Pure and Applied Logic ,
57, no. 1, 1–26.
 Bunge, M., 1974, "Topos Theory and Souslin's Hypothesis",
Journal of Pure and Applied Algebra, 4,
159–187.
 Freyd, P., 1980, "The Axiom of Choice", Journal of Pure and
Applied Algebra, 19, 103–125.
 Freyd, P., 1987, "Choice and WellOrdering", Annals of Pure and
Applied Logic, 35, 2, 149–166.
 Joyal, A. & Moerdijk, I., 1995, Algebraic Set Theory,
Cambridge: Cambridge University Press.
 Scedrov, A., 1984, Forcing and Classifying Topoi,
Providence: AMS.
 Tierney, M., 1972, "Sheaf Theory and the Continuum Hypothesis",
Toposes, Algebraic Geometry and Logic, F.W. Lawvere (ed.),
Springer Lecture Notes in Mathematics 274, 13–42.
D.6.4 Proof theory
 Blute, R. & Scott, P., 2004, "Category Theory for Linear
Logicians", in Linear Logic in Computer Science, T. Ehrhard,
P. Ruet, JY. Girard, P. Scott, eds., Cambridge: Cambridge University
Press, 1–52.
 Cockett, J. R. B. & Seely,
R. A. G., 2001, "Finite Sumproduct Logic", Theory
and Applications of Categories (electronic), 8,
63–99.
 Hyland, J. M. E., 1982, "The Effective Topos",
Studies in Logic and the Foundations of Mathematics,
110, Amsterdam: North Holland, 165–216.
 Hyland, J. M. E., 1988, "A Small Complete Category",
Annals of Pure and Applied Logic, 40, no. 2,
135–165.
 Hyland, J. M. E., 1991, "First Steps in Synthetic
Domain Theory", Category Theory (Como 1990), Lecture Notes in
Mathematics, 1488, Berlin: Springer,
131–156.
 Hyland, J. M. E., 2002, "Proof Theory in the
Abstract", Annals of Pure and Applied Logic,
114, no. 1–3, 43–78.
 Hyland, J. M. E. & Robinson, E. P. &
Rosolini, G., 1990, "The Discrete Objects in the Effective Topos",
Proceedings of the London Mathematical Society (3),
60, no. 1, 1–36.
 Lambek, J., 1986, "Cartesian Closed Categories and Typed lambda
calculi", Combinators and Functional Programming Languages,
Lecture Notes in Computer Science, 242, Berlin:
Springer, 136–175.
 Lambek, J., 1989, "On Some Connections Between Logic and Category
Theory", Studia Logica, 48, 3,
269–278.
 Lambek, J., 1994, "Some Aspects of Categorical Logic", Logic,
Methodology and Philosophy of Science IX, Studies in Logic and the
Foundations of Mathematics 134, Amsterdam: North
Holland, 69–89.
D.6.5 Modal logic and other logical systems
 Freyd, P., 2002, "Cartesian Logic", Theoretical Computer
Science, 278, no. 1–2, 3–21.
 Ghilardi, S., 1989, "Presheaf Semantics and Independence Results
for some Nonclassical firstorder logics", Archive for
Mathematical Logic, 29, no. 2,
125–136.
 Lambek, J., 1993, "Logic without Structural Rules",
Substructural Logics, Studies in Logic and Computation,
2, Oxford: Oxford University Press,
179–206.
 Mc Larty, C., 1986, "Left Exact Logic", Journal of Pure
and Applied Algebra, 41, no. 1, 63–66.
 Rodabaugh, S. E. & Klement, E. P., eds.,
Topological and Algebraic Structures in Fuzzy Sets: A Handbook of
Recent Developments in the Mathematics of Fuzzy Sets, Trends in
Logic, 20, Dordrecht: Kluwer.
 Reyes, G., 1991, "A Topostheoretic Approach to Reference and
Modality", Notre Dame Journal of Formal Logic,
32, no. 3, 359–391.
 Reyes, G. & Zawadowski, M., 1993, "Formal Systems for Modal
Operators on Locales", Studia Logica, 52, no.
4, 595–613.
 Reyes, G. & Zolfaghari, H., 1991, "Topostheoretic Approaches
to Modality", Category Theory (Como 1990), Lecture Notes in
Mathematics, 1488, Berlin: Springer,
359–378.
 Reyes, G. & Zolfaghari, H., 1996, "BiHeyting Algebras, Toposes
and Modalities", Journal of Philosophical Logic,
25, no. 1, 25–43.
E. Applications of category theory
E.1 Books and papers on synthetic differential geometry
 Bell, J. L., 1998, A Primer of Infinitesimal
Analysis, Cambridge: Cambridge University Press.
 Bell, J. L., 2001, "The Continuum in Smooth Infinitesimal
Analysis", Reuniting the Antipodes – Constructive and
Nonstandard Views on the Continuum, Synthese Library,
306, Dordrecht: Kluwer, 19–24.
 Kock, A., 1981, Synthetic Differential Geometry, London
Mathematical Society Lecture Note Series, 51,
Cambridge: Cambridge University Press.
 Lawvere, F. W., 2002, "Categorical Algebra for Continuum
Micro Physics", Journal of Pure and Applied Algebra,
175, no. 1–3, 267–287.
 Moerdijk, I. & Reyes, G., 1991, Models for Smooth
Infinitesimal Analysis, New York: SpringerVerlag.
E.2 Other applications of category theory
 ArziGonczaworski, Z., 1999, "Perceive This as That —
Analogies, Artificial Perception, and Category Theory", Annals of
Mathematics and Artificial Intelligence, 26, no.
1, 215–252.
 Baianu, I. C., 1987, "Computer Models and Automata Theory in
Biology and Medecine", in Witten, Matthew, Eds. Mathematical
Modelling, Vol. 7, 1986, chapter 11, Pergamon Press, Ltd.,
1513–1577.
 Ehresmann, A. C. & Vanbremeersch, JP., 1987,
"Hierarchical Evolutive Systems: a Mathematical Model for Complex
Systems", Bulletin of Mathematical Biology,
49, no. 1, 13–50.
 Healy, M. J., 2000, "Category Theory Applied to Neural
Modeling and Graphical Representations", Proceedings of the
IEEEINNSENNS International Joint Conference on Neural Networks:
IJCNN200, Como, vol. 3, M. Gori, SI. Amari, C. L. Giles,
V. Piuri, eds., IEEE Computer Science Press, 35–40.
F. Major books and papers of the history of category theory
F.1 The first period: the origins and first applications
19421957
 Eilenberg, S. & Cartan, H., 1956, Homological Algebra,
Princeton: Princeton University Press.
 Eilenberg, S. & Mac Lane, S., 1942, "Group Extensions
and Homology", Annals of Mathematics, 43,
757–831.
 Eilenberg, S. & Mac Lane, S., 1945, "General Theory of
Natural Equivalences", Transactions of the American Mathematical
Society, 58, 231–294.
 Eilenberg, S. & Steenrod, N., 1952, Foundations of
Algebraic Topology, Princeton: Princeton University Press.
 Mac Lane, S., 1950, "Dualities for Groups", Bulletin of
the American Mathematical Society, 56,
485–516.
F.2 The second period: the emergence of an autonomous field
19571970
 Dieudonné, J. & Grothendieck, A., 1960, [1971],
Éléments de Géométrie
Algébrique, Berlin: SpringerVerlag.
 Freyd, P., 1964, Abelian Categories. An Introduction to the
Theory of Functors, New York: Harper & Row.
 Grothendieck, A., 1957, "Sur Quelques Points d'algèbre
homologique", Tohoku Mathematics Journal, 9,
119–221.
 Grothendieck, A. et al., Séminaire de
Géométrie Algébrique, Vol. 17, Berlin:
SpringerVerlag.
 Kan, D. M., 1958, "Adjoint Functors", Transactions of
the American Mathematical Society, 87,
294–329.
F.3 The birth of categorical logic
 Freyd, P., 1965, "The Theories of Functors and Models".
Theories of Models, Amsterdam: North Holland,
107–120.
 Freyd, P., 1972, "Aspects of Topoi", Bulletin of the Australian
Mathematical Society, 7, 1–76.
 Lambek, J., 1968, "Deductive Systems and Categories I. Syntactic
Calculus and Residuated Categories", Mathematical Systems
Theory, 2, 287–318.
 Lambek, J., 1969, "Deductive Systems and Categories II. Standard
Constructions and Closed Categories", Category Theory, Homology
Theory and their Applications I, Berlin: Springer,
76–122.
 Lambek, J., 1972, "Deductive Systems and Categories III. Cartesian
Closed Categories, Intuitionâ‰ istic Propositional
Calculus, and Combinatory Logic", Toposes, Algebraic Geometry and
Logic, Lecture Notes in Mathematics, 274, Berlin:
Springer, 57–82.
 Lawvere, F.W., 1963, "Functorial Semantics of Algebraic Theories",
Proceedings of the National Academy of Sciences U.S.A.,
50, 869–872.
 Lawvere, F. W., 1964, "An Elementary Theory of the Category
of Sets", Proceedings of the National Academy of Sciences
U.S.A., 52, 1506–1511.
 Lawvere, F. W., 1965, "Algebraic Theories, Algebraic
Categories, and Algebraic Functors", Theory of Models,
Amsterdam: North Holland, 413–418.
 Lawvere, F. W., 1966, "The Category of Categories as a
Foundation for Mathematics", Proceedings of the Conference on
Categorical Algebra, La Jolla, New York: SpringerVerlag,
1–21.
 Lawvere, F. W., 1969a, "Diagonal Arguments and Cartesian
Closed Categories", Category Theory, Homology Theory, and their
Applications II, Berlin: Springer, 134–145.
 Lawvere, F. W., 1969b, "Adjointness in Foundations",
Dialectica, 23, 281–295.
 Lawvere, F. W., 1970, "Equality in Hyper doctrines and
Comprehension Schema as an Adjoint Functor", Applications of
Categorical Algebra, Providence: AMS, 114.
 Lawvere, F. W., 1971, "Quantifiers and Sheaves", Actes
du Congrès International des Mathématiciens, Tome 1,
Paris: GauthierVillars, 329–334.
 Lawvere, F. W., 1972, "Introduction", Toposes, Algebraic
Geometry and Logic, Lecture Notes in Mathematics, 274,
SpringerVerlag, 1–12.
 Lawvere, F. W., 1975, "Continuously Variable Sets: Algebraic
Geometry = Geometric Logic", Proceedings of the Logic Colloquium
Bristol 1973, Amsterdam: North Holland, 135–153.
 Lawvere, F. W., 1976, "Variable Quantities and Variable
Structures in Topoi", Algebra, Topology, and Category Theory,
New York: Academic Press, 101–131.
 Reyes, G., 1974, "From Sheaves to Logic", in Studies in
Algebraic Logic, A. Daigneault, ed., Providence: AMS.
F.4 On the history of category theory and categorical logic
 Awodey, S. & Reck, E. R., 2002, "Completeness and
Categoricity I. NineteenCentury Axiomatics to TwentiethCentury
Metalogic", History and Philosophy of Logic,
23, 1, 1–30.
 Awodey, S. & Reck, E. R., 2002, "Completeness and
Categoricity II. TwentiethCentury Metalogic to TwentyfirstCentury
Semantics", History and Philosophy of Logic,
23, 2, 77–94.
 Johnstone, P. T., 2001, "Elements of the History of Locale
Theory", Handbook of the History of General Topology, Vol. 3,
Dordrecht: Kluwer, 835–851.
 Lawvere, F. W., 2000, "Comments on the Development of Topos
Theory", Development of Mathematics 19502000, Basel:
Birkhäuser, 715–734.
 Mac Lane, S., 1988, "Concepts and Categories in
Perspective", A Century of Mathematics in America, Part I,
Providence: AMS, 323–365.
 Mac Lane, S., 1989, "The Development of Mathematical Ideas
by Collision: the Case of Categories and Topos Theory", Categorical
Topology and its Relation to Analysis, Algebra and Combinatorics,
Teaneck: World Scientific, 1–9.
 Mc Larty, C., 1990, "Uses and Abuses of the History of Topos
Theory", British Journal for the Philosophy of Science,
41, 351–375.
 Van Oosten, J., 2002, "Realizability: a Historical Essay",
Mathematical Structures in Computer Science,
12, no. 3, 239–263.
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Stanford Encyclopedia of Philosophy