Category Theory for All (NASSLLI 2012)

Category Theory for All:
a logician's perspective

Valeria de Paiva

NASSLLI2012

June, 2012

Valeria de Paiva (NASSLLI2012) June, 2012 1 / 113

Outline

Outline

1 Interested in Category Theory?

2 Categories, Functors and Natural Transformations

3 Adjunctions

4 Deductive Systems as Categories

5 A taste of Glue Semantics?...


Interested in Category Theory?

Introduction
Category theory is a relatively new branch of mathematics that keeps
appearing in texts of computational semantics, on the syntax-semantics
interface and even on material on pure syntax.
It also is very proeminent on the literature on Functional Programming, in
the theory of specications and even on some discussions on databases.
Thirdly it has become familiar in texts for physicists, especially the ones
interested in Quantum stu.

Category theory is simple, but requires an investment of time to get over
the initial barrier of language, hence it seems an ideal subject to be taught
at NASSLLI.

There are two ways to do great mathematics. The rst way is to
be smarter than everybody else. The second way is to be stupider
than everybody else but persistent. Raoul Bott



A little bit of personal history...

Some twenty years ago I nished my PhD thesis The Dialectica
Categories in Cambridge. My supervisor was Dr Martin Hyland.



Dialectica categories come from Gödel's Dialectica
Interpretation

The interpretation is named after the Swiss journal Dialectica where it
appeared in a special volume dedicated to Paul Bernays 70th birthday in
1958.
I was originally trying to provide an internal categorical model of the
Dialectica Interpretation. The categories I came up with proved (also) to
be a model of Linear Logic...



Dialectica categories are models of Linear Logic

Linear Logic (LL) introduced by Girard (1987): [...]linear logic comes from
a proof-theoretic analysis of usual logic.
LL the best of both worlds, the dualities of classical logic plus the
constructive content of proofs of intuitionistic logic.

Dialectica categories are a cool model of LL, still one of the best around...



From Linear Logic to Semantics of Natural Language

Many linguists became interested in Linear Logic because it's a
resource-conscious logic.
I became interested in Linguistics because of Linear Logic and its
programme of proofs as rst class objects.



From Linear Logic to Linear Functional Programming

I also became interested in Functional Programing because of Linear Logic
and its programme of proofs as rst class objects.



Proofs as rst class objects?

Programme to elevate the status of proofs to rst class logical objects.
Instead of asking `when is a formula A true', we ask `what is a proof of A?'
Using Frege's distinction between sense and denotation:
proofs are the senses of logical formulas,
whose denotations might be truth values.

Sometimes I call this programme Proof Semantics.
sometimes I call it Categorical Proof Theory (because the semantics of proofs
are given in terms of natural constructions in Category Theory).



The Challenges of Proof Semantics...

There is a problem with viewing proofs as logical objects:
We do not have direct access to proofs.
Only to syntactic representations of them, in the form of derivations in
some proof system (e.g. axiomatic, natural deduction or sequent calculus).

Syntactic derivations are awed means of accessing the underlying proof
objects. The syntax can introduce spurious dierences between derivations
that do not correspond to dierences in the underlying proofs; it can also
mask dierences that really are there.
Semantics of proofs is what categorical logic is about.
Our category theory will be directed towards proof semantics.


Things I hope you know...

We will assume some familiarity with rst-order predicate calculus, a
rudimentary knowledge of the lambda-calculus and type theory, and a basic
idea of what logical inference is about.
If you have an intuitive understanding what the following means I think
you'll be ne.

∀x. man(x) → mortal(x), man(john) mortal(john)
λx.see(x, john) (f red) ≡β see(f red, john)

If you have done a basic course in abstract algebra, linear algebra or
groups, then you should be in really good shape.



What is Categorical Logic?

Traditionally mathematical logic divided into four areas:
proof theory
model theory,
recursion theory
and set theory.
A trend towards adding `Complexity' to the four basic areas.

Categorical logic means dierent things to dierent people.
In this course we will mean categorical `proof theory'.



What is Categorical Proof Theory?

Proof theory using categorical models instead of Sets-based models...
Logic could be rst-order, but it will be propositional only.
Classical logic poses some problems, discussed later on.
We deal with propositional intuitionistic logic IPL.

For this course this means that we need to consider three independent
connectives: conjunction (A ∧ B ), disjunction (A ∨ B) and implication
(A → B ).
These are not interdenable, so some of the usual classical rules (e.g.
A → B ∼ ¬A ∨ B )
= are not valid.



The quest for proofs...

Traditional proof theory, to the extent that it relies on models uses
algebraic structures such as Boolean algebras, Heyting algebras or Kripke
models of several styles.
These models lose one important dimension. In these models dierent
proofs are not represented at all.
Provability, the fact that Γ a collection of premisses A1 , . . . , A k entails A,
is represented by the less or equal ≤ relation in the model.
This does not give us a way of representing the proofs themselves. We only
know if a proof exists Γ≤A or not. All proofs are collapsed into the
existence of this relation.


The quest for proofs...

By contrast in categorical proof theory we think and write a proof as

Γ →f A

where f is the reason why we can deduce A from Γ, a name for the proof
we are thinking of. Thus we can observe and name and compare dierent
derivations. Which means that we can see subtle dierences in the logics.



Constructive perspective: not much of one...
This course is not about the deep philosophical questions of the foundations
of mathematics. We will side-step every philosophical issue that we can.
(It is not that these issues are not interesting. They are. It is just that I
don't think I can enlighten anyone much on them.)

Caveat emptor Whether you believe or not that there is a single logic that
is the correct one, applications (especially in CS) have provided us with
empirical evidence that there are several logical systems that are convenient
for use in specic contexts.
When studying these contexts and applications it pays to keep a
constructive viewpoint. If classical logic turns out to be necessary, we will
pragmatically use it and ag its use.
Commmitment to intuitionistic principles is a matter of pragmatic choice:
It is easier to be intuitionistic and then if need arises to become classic
than the other way round. Moreover it's safer: intuitionistic principles are
accepted by all...



The real point of this course...



Edwardian proofs for futuristic programs...



Introductions to category theory?...

There are many. check out logic-forall for links.
S. Mac Lane, Categories for the Working Mathematician (1971, 1998)
J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types (1989)
M. Barr, C. Wells, Category Theory for Computer Science (1995)
B.C. Pierce, Basic Category Theory for Computer Science (1991)
R. Blute P. Scott, Category Theory for Linear Logicians (2003)
F.W. Lawvere, S.H. Schanuel, Conceptual Mathematics (1997)
Abramsky,Tzevelekos, Introduction to Cats and Categorical Logic (2010)
S. Awodey, Category Theory, (2010).
H. Simmons, An Introduction to Category Theory, (2011)
V. de Paiva, Categorical Proof Theory and Linear Logic (1996)

Blog http://logic-forall.blogspot.com/



More Introductions...

M. Johansson, Category Theory and Functional Programming, Stanford
FALL 2009, and St. Andrews 2012
also Chalmers discussion group:
G. Hutton, Introduction to Category Theory , Midlands Grad School, 2012
D. Verity, An Introduction to Category Theory - FP-Syd talk,
E. Cheng, The Catstars YouTube channel,
J. van Oosten, Basic Category Theory, 2002,
L. Maertens, Category Theory for Program Construction. ESSLLI (1995)
Benton, Bierman, Hyland and de Paiva, Term Assignment for Intuitionistic
Linear Logic, 1992.



Elements of category theory...

Meant to imply that we will discuss and do exercises on some basic
concepts. necessary for categorical proof theory, in the restricted sense
above.
No promisses of completeness of building blocks...
Basic concepts we intend to cover:
categories, functors, natural transformations
products, pullbacks, equalizers, initial and terminal objects
adjunctions and monoidal closure
Yoneda Lemma
(monads, algebras, coalgebras and comonads?)


Categories, Functors and Natural Transformations

Denition of a Category

A category C consists of a collection of objects (A, B, . . .) and a collection
of morphisms (or maps) from A to B (written asf : A → B ) satisfying
three easy conditions:

1. Given maps f : A → B and g : B → C such that the codomain of f is
equal to the domain of g , there is an operation of composition of
morphisms, which produces a map f ; g : A → C .
2. The composition operation is associative, ie. f ; (g; h) = (f ; g); h.
3. For each object A in the category we have an associated identity
morphism idA : A → A such that if f : A → B is any map in the category,
then idA ; f = f = f ; idB .

Note on `writing' composition: mathematicians usually write g◦f for what
I mostly write as f ; g. Use whatever you like, and have a scratchpad nearby
to check it out...



The Denition of Category



Examples of Categories



Examples of kinds of Categories




The category of Sets and usual functions is the paradigmatic example.
Maybe it's too intuitive.

A good example to show the dierence between morphisms and functions is
the category Rel of sets and relations.

Easy mathematical examples:
Groups and homomorphisms,
vector spaces and linear transformations,
monoids and monoid homomorphisms.
More complicated example: topological spaces and continuous maps.



Examples of Categories: Monoids

A monoid M is a set with a `multiplication-like' operation and one identity,
say the element e. So M = (M, ·, e) where ·: M × M → M is such that

m1 · (m2 · m3 ) = (m1 · m2 ) · m3 and m· e = m = e · m

Easy mathematical examples:
(N, +, 0) integers under addition,
(R, ×, 1) reals under multiplication
matrices under multiplication of matrices (Matn , ·, In )
strings over alphabet {0, 1}, under concatenation, (String, ·, λ).



Two extreme examples

The empty category (writen as 0), which has no objects nor morphisms.
(Just a parallel concept to the empty set.)
The singleton category (written as 1), which has a single object and a
single morphism, which is the identity on the single object.



Why Categories?

Category theory (CT) is a convenient new language
It puts existing mathematical results into perspective
It gives an appreciation of the unity of modern mathematics

As a language, it oers economy of thought and expression
Reveals common ideas in (apparently) unrelated areas of mathematics
A single result proved in CT generates many results in dierent areas
Duality: for every categorical construct, there is a dual, reverse all the maps
Dicult problems in some areas of mathematics can be translated into
(easier) problems in other areas (e.g. by using functors, which map from
one category to another)
Makes precise some notions that were previously vague, e.g. universality,
naturality, etc



Why Categories?
Category theory is ideal for:
Reasoning about structure and the mappings that preserve structure
Abstracting away from details
Automation (constructive methods for many useful categorical structures)

Applications of Category theory in software engineering:
The category of algebraic specications: category theory used to represent
composition and renement
The category of temporal logic specications: category theory used to build
modular specications and decompose system properties across them
Automata theory: category theory oers a new way of comparing automata

Logic as a category: represent a logical system as a category, and construct
proofs using universal constructs in category theory (diagram chasing).
A category of logics: theorem provers in dierent logic systems can be
hooked together through institution morphisms
Functional Programming: type theory, programming language semantics...


Beware of paradoxes...

To avoid set theoretical paradoxes care must be taken.
Oversimplifying: small collections are sets, big collections are classes and
try not to pay much attention to the dierences between them.

Say C is a small category if both
the collection objects of C (written as obj(C) or |C|) and
the collection of maps between two objects HomC (A, B) (for all A, B
objects of C) are sets.
Say C is a locally small category if HomC (A, B) is a set for any two
objects A and B in the category.

We assume all the categories we discuss are at least locally small and that
the sets of morphisms HomC (A, B) are disjoint for distinct pairs of objects
A and B. We also draw diagrams and say they commute to state
equations between morphisms.



beginframe



Initial and Terminal Objects



Recap

1. Discussed the idea of categorical semantics:



Recap

want our to be able to distinguish dierent derivations of the same formula
from same assumptions;



Recap

2. want to use categorical morphisms as denotations of our proofs;



Recap

3. want Curry-Howard triangles for other logics/other type
theories/other categories;



Recap

for now want to understand how original categorical Curry-Howard works



Recap

for now want to understand how original categorical Curry-Howard works
4. Saw the denition of a category and some examples



Repeating: a Category

A category C consists of a collection of objects (A, B, . . .) and a collection
of morphisms (or maps/arrows) from A to B (written as f : A → B)
satisfying three easy conditions:

1. Given maps f : A → B and g : B → C such that the codomain of f is
equal to the domain of g , there is an operation of composition of
morphisms, which produces a map f ; g : A → C .
2. The composition operation is associative, ie. f ; (g; h) = (f ; g); h.
3. For each object A in the category we have an associated identity
morphism idA : A → A such that if f : A → B is any map in the category,
then idA ; f = f = f ; idB .

Note on `writing' composition: mathematicians usually write g◦f for what
I mostly write as f ; g. Use whatever you like, and have a scratchpad nearby
to check it out...



Repeating: Examples

0. Category Set, plus few nite ones, plus `sets-with-structure' (universal
algebra)...

1.any partially-ordered set (P, ≤) can be seen as a category, a morphism
exists between p, q in P just when p ≤ q, ie. either a single or no morphism
between objects.



Repeating: Examples

0. Category Set, plus few nite ones, plus `sets-with-structure' (universal
algebra)...

1.any partially-ordered set (P, ≤) can be seen as a category, a morphism
exists between p, q in P just when p ≤ q, ie. either a single or no morphism
between objects.

how do we show this satises conditions for category?
2. category of monoids (M, ·, e) and monoid homormorphisms, ie maps
f :M →M such that f (m · n) = f (m) · f (n) and f (e) = e .



Two extreme examples of categories


single morphism,



Two extreme examples of categories


single morphism, which is the identity on the single object.



The opposite category Cop

Given any category C we can formally invert all its morphisms, to construct
the category Cop .
Why would you do this?

Actually will be extremely useful to deal with e.g. logical implication.
But for the time being, it's a simple way of seeing that morphisms do not
have to be function-based.
Fork on the road: structures inside categories vs. structures relating cats



Products of Categories

Given categories C and D we consider a new category whose objects are
pairs of objects (C, D), C in C and D in D and morphisms are pairs of
morphisms (f, g), f : C → C , g : D → D . This builds the category
C × D.



Products of Categories

Given categories C and D we consider a new category whose objects are
pairs of objects (C, D), C in C and D in D and morphisms are pairs of
morphisms (f, g), f : C → C , g : D → D . This builds the category
C × D.
Does the operation of taking products of categories has a unity?



Slice Categories

This construction is a bit more involved. Given a category C and an object
A of C, the slice category over A, C/A is a category whose objects are
morphisms of the form C → A.
A morphism from C → A to C → A is a morphism in C, f : C → C such
that the following triangle commutes:

f -
C C

?
A



A menagerie of Categories



Functors

We have been stressing that more than objects it is the way that objects
relate to each that matters in category theory. The same is true of
categories themselves.
Hence the old Eilenberg-MacLane joke that categories were only invented
so that they could talk about functors. And functors were only invented to
talk about natural transformations. and yes talk about the natural
transformations was what the algebraic geometers wanted to do all along.
So we need maps of categories, the natural way of relating categories.
Which structure should they preserve?



Functors

We have been stressing that more than objects it is the way that objects
relate to each that matters in category theory. The same is true of
categories themselves.
Hence the old Eilenberg-MacLane joke that categories were only invented
so that they could talk about functors. And functors were only invented to
talk about natural transformations. and yes talk about the natural
transformations was what the algebraic geometers wanted to do all along.
So we need maps of categories, the natural way of relating categories.
Which structure should they preserve?

Composition and identities, of course.


Functors

Given categories C and D, F : C → D consists
a functor of an assignment
of objects of C to objects to D such that if f : A → B is a morphism in C
then F (f ) : F (A) → F (B) is a morphism in D and
1. F preserves composition, F (f ; g) = F (f ); F (g) and
2. F preserves identities, F (idA ) = idF (A)



Functors



Examples of Functors

The trivial example is the identity functor on any category C.
1. A functor between posets is simply an order-preserving function.
(they are called monotonic or monotone functions..)
2. Forgetful functors simply forget the mathematical structure present. For
example, the functor U : Mon → Sets simply forgets the monoid structure
and returns the set that is the carrier of the monoid.
Similarly for U : Group → Sets, the functor that forgets the group
structure.
3. More interesting are the free functors, that create free structure on bare
sets. For example, FreeMon : Sets → Mon creates free monoids.



Examples of Functors

The trivial example is the identity functor on any category C.
1. A functor between posets is simply an order-preserving function.
(they are called monotonic or monotone functions..)
2. Forgetful functors simply forget the mathematical structure present. For
example, the functor U : Mon → Sets simply forgets the monoid structure
and returns the set that is the carrier of the monoid.
Similarly for U : Group → Sets, the functor that forgets the group
structure.
3. More interesting are the free functors, that create free structure on bare
sets. For example, FreeMon : Sets → Mon creates free monoids.

Can you guess what the free monoid functor FreeMon does to a set
A = {a, b, c} ?



More Slice Categories
Given a map f: A→B in C we have a functor Σf : C/A → C/B .
For g : C/A → C/A
g -
C C

?
A



More Slice Categories
Given a map f: A→B in C we have a functor Σf : C/A → C/B .
For g : C/A → C/A
g -
C C

?
A
Σf acts as
g -
C C
-

? ?
A - B



More Examples of Functors
From Hutton's notes, http://www.cs.nott.ac.uk/ gmh/cat.html.
1. The powerset functor, P : Set → Set, P(A) is the set of all subsets of
A, for example if A = {0, 1} then P(A) = {∅, {0}, {1}, {0, 1}}.
2. The List : Set → Set functor. On objects ListA is the set dened by the
grammar
ListA ::= Nil | ConsA(ListA)
On maps, if f: A→B is a function then List(f ) : List(A) → List(B) is
the function dened by the equations:



grammar

List f (Nil) = Nil



grammar

List f (Nil) = Nil

List f (Cons x xs) = Cons(f (x))(List f xs))



grammar

List f (Nil) = Nil


Preservation of composition and identities proved by induction.
3. Can you do the functor Tree : Set → Set such that TreeA is the set of
binary trees whose leaves are elements of the set A?


Recap:

1. Discussed categories. what are they?



Recap:

1. Discussed categories. what are they? collection of objects, collection of
maps such that maps compose (associatively) and there are identities for
each object
f ; (g; h) = (f ; g); h
f (idA ) = idf (A)
2. talked about some examples e.g. monoids and monoid morphisms



Recap:

each object
f ; (g; h) = (f ; g); h
f (idA ) = idf (A)
can you remember two examples of monoids?
slice category where objects are already morphisms..
product category C × D.
3. dened functor F : C → D need to map objects to objects and maps to
maps. need to satisfy F (f ; g) = F (f ); F (g) F (idA ) = idF (A)



Recap:

each object
f ; (g; h) = (f ; g); h
f (idA ) = idf (A)
can you remember two examples of monoids?
slice category where objects are already morphisms..
product category C × D.
3. dened functor F : C → D need to map objects to objects and maps to
maps. need to satisfy F (f ; g) = F (f ); F (g) F (idA ) = idF (A) where are
these compositions?



Two more examples of Categories: I. Labelled Graphs

Can you describe the category of `labelled graphs' ? what are the objects?



Labelled Graphs

Morphisms/arrows/maps from one labelled graph to another?



Maps of Labelled Graphs



Second Example: Automata

remember these?



Second Example: Automata from Goguen's Manifesto




Back to the powerset functor, P : Set → Set, P(A) is the set of all subsets
of A, for example if A = {0, 1} then P(A) = {∅, {0}, {1}, {0, 1}}.
if B = {a, b, c} then
P(B) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
You can also consider only nite subsets of a set Pf in (A), which makes
more sense if the set is innite.

The power set functorP : Set → Set maps each set to its power set and
each function f : X → Y to the map which sends U ⊆ X to its image
f (U ) ⊆ Y . One can also consider the contravariant power set functor
which sends f :X→Y to the map which sends V ⊆Y to its inverse
−1
image f (V ) ⊆ X .




The List : Set → Set functor. On objects ListA is the set dened by the
grammar




grammar

List f (Nil) = Nil




grammar

List f (Nil) = Nil


Preservation of composition and identities proved by induction.
3. Can you do the functor Tree : Set → Set such that TreeA is the set of
binary trees whose leaves are elements of the set A?



Back to math: Hom Functors

Pick a category C and an object A in it. Then HomC (A, −) is a functor
from C to Set, where HomC (A, −) the collection of all morphisms of C
with source A.

F is
A functor represented by an object A if F is naturally isomorphic to
HomC (A, −).



More Hom Functors

Pick a category C and an object A in it. Look at HomC (−, A) is a functor
from C to Set, where HomC (−, A) is the collection of all morphisms of C
with codomain A.



Functors: more notation

A functor of the form H : C × D → E, where the domain is a product
category, is called a bi-functor.
Projections from a product category π1,2 : A × B → A are bi-functors.
op
A functor of the form F: C → D, whose domain is the opposite of a
given category, is called a contravariant functor.

Example of non-functor that looks like one: the centre of a group (the
center Z(G) of a group G, is dened as the subgroup consisting of all
elements g ∈ G such that for all elements h ∈ H the equality gh = hg
holds).



Functors are ubiquitous?

Graham says:

The wikipedia says: For functors as a synonym of function objects in
computer programming to pass function pointers along with its state, see
function object. For the use of the functor morphism presented here in
functional programming see also the fmap function of Haskell's Functor
class. Functor is also used to name functions in Prolog. For functors in
linguistics, see Function word.



Natural Transformations
Given functors F : C → D and G : C → D, a natural transformation
α : F → G consists of morphisms αA : F (A) → G(A) such that, for any A
in C and for any f : A → B in C we have:

αA-
F (A) G(A)

F (f ) G(f )
? ?
F (B) - G(B)
αB

do easy case and mention
http://blog.sigfpe.com/2008/05/you-could-have-dened-natural.html
`Starting with the idea of a function that doesn't `peek inside' its
arguments, we're led inexorably to the idea of substitutability of arguments
and from there to a categorical denition of a natural transformation.'


... prettier picture from Edsko de Vries, thanks!



Examples of Natural Transformations

Reversing lists. If A is a set, then revA : List → List is a function dened
by:
revA (Nil) = Nil
revA (Cons x xs) = revA (xs) · (Consx Nil)
To show this is natural must show

revA
ListA - ListA

Listf Listf
? ?
ListB - ListB
revB



Constructions in Categories: Products

A category C A and B in C, there
has binary products, if given objects
exists an objectA × B and morphisms π1 : A × B → A and
p2 : A × B → B of C such that if P is another object of C equipped with
morphisms p1 : P → A and p2 : P → B , then there's a unique map
u : P → A × B such that the triangles below commute.

P

p2
p1

u

-

π1 ? π2 -
A A×B B



Constructions in Categories: Pullbacks

A category C f : X → Z and g : Y → Z in C,
has pullbacks, if given maps
there exists an object P equipped with morphisms p1 : P → X and
p2 : P → Y of C such that if Q is another object of C equipped with
morphisms q1 : Q → X and q2 : Q → Y , then there's a unique map
u : Q → P such that the `triangles' below commute.



Constructions in Categories: Equalizers

A category C has equalizers, if given maps f: A→B and g : A → B in
C, there exists an object E equipped with a morphism e : E → A of C
such that if E is another object of C equipped with a morphism
e : E → A, then there's a unique map e: E → E such that the triangle
below commutes.



Recap...



Recap...

Got up our ladder: categories, functors and natural transformations.
Cats have objects and maps, identity morphism for each object, satisfying
composition and identity law.
Functors transform cats into other cats,F : C → D have a component for
objects a component for maps, st F (f ; g) = F (f ); F (g) (covariant
functor) and F (idC ) = idF C .
constructions relating cats: opposite or dual cat, products, extra examples
like slice cat and Hom-functor. constructions inside a cat: initial/terminal
object, products (coproducts), pullbacks (push-outs), equalizers
(co-equalizers).
Not worrying about size considerations (set-theoretic paradoxes) can build:
1. category of small categories where morphisms are functors between cats.
2. Cat of functors between two given cats F un(C, D) where objects are
functors and morphisms are natural transformations.



Recap...

A nat transformation α: F → G between functors F and G both from C
to D is a collection of morphisms in D αC one for each object C of the
category C satisfying a nice square, for each morphism in C f: C →C :

αC-
F (C) G(C)

F (f ) G(f )
? ?
F (C ) - G(C )
αC
only example reverse between lists in Set...



Trivial example: identity natural transformation

α − idC : F → F .
how does it go?



Trivial example: identity natural transformation

α − idC : F → F .
how does it go?
A nat transformation α: F → G between functors F and G both from C
to C is a collection of morphisms in C αC one for each object C of the
category satisfying a nice square, for each morphism in C f: C →C :

α − idC
F (C) - F (C)

F (f ) F (f )
? ?
F (C ) - F (C )
α − idC


Adjunctions

Adjunctions?

Given functors F: C →D and G : D → C, we say that F is left-adjoint to
G (or that G is right-adjoint to F ) F G if there is a natural bijection

HomD (F C, D) ∼ HomC (C, GD)
=

for each pair of objects C in C and D in D.

Many people prefer to think of them as Galois connections.


Adjunctions

Galois connections...

Let (A, ≤) and (B, ≤) be two partially ordered sets. A Galois connection
between these posets consists of two monotone functions: F: A→B and
G : B → A, such that for all a in A andb in B, we have

F (a) ≤ b if and only if a ≤ G(b)
In this situation,F is called the lower adjoint of G and G is called the
upper adjoint of F.
Several names in the literature, wikipedia has a good selection.
Originally from Galois' work on groups.
Check out Peter Smith's carefully written The Galois Connection between
Syntax and Semantics.
Abstract Algebraic Logic. Abstract interpretation. Concept analysis,
some Information Flow...


Adjunctions

Examples of Adjunctions

Free monoids again... Cartesian Closedness (monoidal closedness)


Adjunctions

Free monoids?

Recall the functor Free : Set → Mon.
On an set A it produces the free monoid. i.e sequences of elements from
A, 'multiplied' by concatenation, (A∗ , [], concat).
On an set B it produces the free monoid (sequences of elements from B)
(B ∗ , [], concat).


Adjunctions

Free monoids?

(B ∗ , [], concat).
remind me, what's the rest of the functor?


Adjunctions

Free monoids?

(B ∗ , [], concat).
remind me, what's the rest of the functor?
On a map f: A→B the action of Free, f ∗ is simply doing the function f
in each of the elements of A.
Recall also the `boring' forgetful functor U : Mon → Set. Given any
monoid (M, ·, e) it returns simply M . Given a monoid morphism
h : (M, ·, e) → (M , ·, e ), return simply the function h.


Adjunctions

Free monoids?

Now let us look at the denition of adjunction: Two functors F, G going in
opposite directions st

=
where ∼
= means there is a bijection that is natural in C and D.


Adjunctions

Free monoids?

Now let us look at the denition of adjunction: Two functors F, G going in
opposite directions st

=
where ∼
= means there is a bijection that is natural in C and D.
We do have functors going between categories Mon and Set in opposite
directions.
U : Mon → Set and Free : Set → Mon
Do we have ....

HomSet (U M, A) ∼ HomMon (M, FreeA)
=
More than 20 dierent denitions of adjunction...


Adjunctions

Free monoids?
Simmons says:There is a lot going on with adjunctions and you will
probably get confused more than once. Don't worry....

universal property: a morphism η : A → U (FreeA) such that whenever M
is a monoid, and g : A → U (M ) a function, then there is a unique monoid
morphism h : Free(A) → M such that U (h) · η = g .

A - UM
-

?
U (Free(A))
Go and read it at your own pace. for intuition think the free monoid is the
monoid built on the generators, the elements of the original set. you can
always embedd the original set into the set that is the forgotten monoid..

Adjunctions

An über familiar adjunction...

HomSet (A × B, C) ∼ HomSet (A, C B )
=
So f: A×B →C f : A → CB.
is bijectively related to
This is trivial currying of functions, if your f has 2 arguments a and b, you
can transform it into a function f of a single argument a that returns a
function taking b's to c's, right?
Schoennkel (1920s) the rst (?) to notice its importance for combinators .
Now can we do it in a generic category C?
We have products. Need to internalize the notion of maps in C from B to
C.
In Set is easy, the set of functions from X to Y may be denoted X→Y or
X
Y .
As a special case, the power set of a set X may be identied with the set
of all functions from X to {0, 1}, denoted 2X .


Adjunctions


Consider the functor − × B : Set → Set, where B is a xed object of Set.
On object A we have A × B ,
on object A we have A × B .


Adjunctions


Consider the functor − × B : Set → Set, where B is a xed object of Set.
On object A we have A × B ,
on object A we have A × B .
what's the morphism component?
On a map f: A→A we simply product with the identity on B.
Now consider the functor (−)B :
→ Set. Set
B
On object D , D is the set of all maps B → D .
On object D it is the set of all maps B → D .
and on a morphism, f : D → D
it's the usual, composition.
Now, if we formally plug in these two functors into our denition of
adjunction, what happens?


Adjunctions


Look at:

=
where F =−×B and G = (−)B


Adjunctions


Look at:

=
where F =−×B and G = (−)B
applying to objects C and D we have:

HomSet (C × B, D) ∼ HomSet (C, DB ) ∼ HomSet (C, B → D)
= =

which corresponds to our trivial operation on of currying functions...
So I want to conclude that in a generic category C with products, I have an
adjunction between functors −×B and (−)B .


Adjunctions


ok, I'm cheating. I didn't tell you how to construct the functor (−)B in
other categories. We know we can do it in Set, but what about anywhere
else?

I can give a proper denition of (−)B if I work hard on limits, the general
notion of which we saw 3 instances: products, pullbacks and equalizers.
then the adjunction is obvious.

or I can give you the adjunction saying you've used it a billion times before
and say, go get your general denition of internalizing functions from the
adjunction. Ie dene function spaces as the functor that makes the
adjunction work, like many books do.
A bit like riding a bike, if you start you can keep your balance...


Adjunctions

Cartesian Closed Categories...

A category C is called a cartesian closed category (CCC) if it has products
and if for each object X of C, the product functor −×X: C →C has a
X
right adjoint, written as (−) . That is we have

HomC (A × X, B) ∼ HomC (A, B X )
=

A minor modication, not insisting on real cartesian products, produces
the denition of symmetric monoidal closed category.

HomC (A ⊗ X, B) ∼ HomC (A, [X, B]C )
=


Adjunctions

Have you seen this adjunction before?...

If you squint a little and make × the same as ∧ and → logical implication
and think of maps in the category as proofs, you can read

HomC (A × X, B) ∼ HomC (A, X → B)
=
as any proof of A∧X →B corresponds bijectively to a proof of
A → X → B, which is a form of the Deduction Theorem in logic.


Deductive Systems as Categories

Lambek introduced the idea of using Gentzen methods in category theory
and linguistics in the Sixties.
A deductive system is simply a labelled graph whose objects are formulas
and whose edges are labelled sequents. There are special identity arrows
and some special rules for creating new sequents outof the old ones. Given
a graph G



What are the Deductive Systems trying to do?...



How do sequents relate to categories?...

Basic idea: a formula corresponds to an object in a suitable category
We assign to each sequent Γ A a map in the suitable category C from
the interpretation of Γ to the interpretation of A. (But we don't bother to
show the dierence between A formula and A object in the category...)
Clearly the interpretation must be compositional: we get the interpretation
of Γ = A1 , . . . , A n from the interpretations of the components Ai .
What is the less contraining construction we can do in a category that will
interpret the context Γ?
People will worry about commutativity, associativity of commas.
If I am talking about intuitionistic logic, where conjunction is commutative
and associative, cartesian product would be ne, but a commutative tensor
product ⊗ is ne too.



How do sequents relate to categories?...

Tensor products? Well, I mentioned categorical products and their universal
property. It turns out that universality is not really required to model
contexts.
So we can do simply with a product-like functor ⊗: C × C → C, which is
associative, commutative and has an identity, usually written as I.
Categorical products are unique up to iso, so if you have products, that's it.
(up to iso).
But tensor products are not unique, you can have several, this corresponds
to several kinds of conjunction in your logic.
A category with an associated tensor product and unit is called a monoidal
category. A category with an associated categorical product and terminal
object is called a cartesian category.
Monoidal categories are very fashionable, as they seem more exible for
modelling phenomena. (glossing over issues with coherence).



How do sequents relate to categories? ...

Adding terms/morphisms to a suitable version of sequent calculus rules we
get a full interpretation of the suimply typed lambda-calculus in categories.
This allows us to think of proofs as λ-terms or of morphisms in the
category build up from identities and the constructors we allowed.
Are we there yet?



How do sequents relate to categories? ...

Adding terms/morphisms to a suitable version of sequent calculus rules we
get a full interpretation of the suimply typed lambda-calculus in categories.
This allows us to think of proofs as λ-terms or of morphisms in the
category build up from identities and the constructors we allowed.
Are we there yet?
No, must prove it works.
Must prove we have triangle...



Old picture



Problems?..

We can add terms to Natural Deduction (Curry-Howard correspondence).

But need sequent formulation provably equivalent to Natural Deduction.

(there is a problem with associating categorical constructions to sequent
calculus rules without ND, the proof equalities don't work without extra
constraints, Herbelin.)
We will discuss this tomorrow.



Recap..

So I goofed, should've dened:
An object C in a category D is an internal hom object or an exponential
object [A → B] or B A if it comes equipped with an arrow
ev : [A → B] × A → B , called the evaluation arrow, such that for any
other arrow f : C × A → B , there is a unique arrow λf : C → [A → B]
such that the composite
C × A →λf ×1A [A → B] × A →ev B
is f .
but yes, the bike metaphor is still valid...



Traditional Curry-Howard reloaded..




But 88 pages in Gallier to prove ND equivalent to sequent calculus...
Once you have it:



Categorical Curry-Howard reloaded..




What can we prove?
Few more (traditional denitions)
A signature is a collection of basic types (Nat, Bool, Char, ...) and typed
functional symbols (add, succ,..., etc).
Consider a basic signature that contains at least basic types 1, 0 and basic
operators × for conjunctions, + for disjunctions and → for logical
implication.
Take a cartesian closed category C with products, coproducts and function
spaces.
An interpretation I for signature Σ in C ( a Σ-structure in C) is specied
by giving an object of C for each type in Σ and a map in C for each
function symbol f in Σ.




Considering terms in context, for any interpretation of Σ in C there exists a
standard inductive denition of the interpretation of terms in context,
induced by the interpretation of Σ.
Consider equations in context. Say an equation in EqΣ holds for an
interpretation I if the interpretations of the terms in context are equal
morphims in the category.
Say an interpretation is a model of a set of equations if it satises all the
equations in the set.
Finally can state:




Soundness Theorem[Categorical Modelling of IPL]
Let C be a cartesian closed category with coproducts and let I be a model
of EqΣ in C. Then I satises any equation derivable from Eqσ using
equational logic.




Soundness Theorem[Categorical Modelling of IPL]
Let C be a cartesian closed category with coproducts and let I be a model
of EqΣ in C. Then I satises any equation derivable from EqΣ using
equational logic.

Completeness Theorem[Categorical Modelling of IPL]
For all signatures Σ there exists a cartesian closed category with coproducts
C and an interpretation of the calculus in this category such that:
If Γ t:A and Γ s : A are derivable, then t and s are interpreted as the
same morphism only if t = s is provable from the equations above using
typed equational logic.



Wadler's Take


A taste of Glue Semantics?...

Glue semantics in 5 min
Thanks for slides, Dick!

Glue semantics is a theory of the syntax-semantics interface that uses
linear logic for meaning composition.
Distinguish two separate logics in semantic interpretation
1. Meaning logic: target logical representation
2. Glue logic: logical specication of how chunks of meaning are assembled

In principle, Glue uses any of several grammar formalisms/any of the
mainstream semantics. In practice Glue started for LFG, with a vanilla
Montague-style logic for meanings.



Linguistic applications of linear logic

1. Categorial and type-logical grammar (Moortgat, vanBenthem):
including parsing categorial grammars (Morrill, Hepple) and
compositional semantics of categorial grammars (Morrill, Carpenter)
2. `Glue semantics' (a version of categorial semantics without an
associated categorial grammar?) (Dalrymple, Lamping Gupta))
3. Resource-based reformulations of other grammatical theories
Minimalism (Retore,Stabler)
Lexical Functional Grammar (Saraswat,Muskens)
Tree Adjoining Grammar (Abrusci)
4. Also AI issues such as the frame problem (White) or planning
(Dixon) with linguistic relevance



Curry-Howard Isomorphism (CHI)

CHI = Pairing of proof rules with operations on proof terms
But doesn't work for all logics, or proof systems

Denes interesting identity criteria for proofs
Syntactically distinct derivations corresponding to same proof

Intimate relation between logic and type-theory.
Varied applications, e.g.
Proofs as programs

Semantic construction for natural language



Example: Input to Semantic Interpretation

Lexicon

Word Meaning Glue
John john ↑ where ↑= g
Fred fred ↑ where ↑= h
saw λy.λx. see(x, y) ↑ .OBJ −◦ (↑ .SU BJ −◦ ↑)
where ↑= f , f.OBJ = h, f.SU BJ = g
Constituents g, h, f : semantic resources, consuming producing meanings



Lexical Premises: Their nature

saw
aλy.λx. see(x, y) : bh −◦ (g −◦ f )

cMeaning Term dGlue Formula
(Propositional LL)

Atomic propositions (f, g, h):
• Correspond to syntactic constituents found in parsing
• Denote resources used in semantic interpretation
(Match production consumption of constituent meanings)
Meaning terms:
• Expressions in some chosen meaning language
• Language must support abstraction and application
• . . . but otherwise relatively free choice



Cutting and Pasting 1...



The Form of Glue Derivations

Γ M:f
where
• Γ is set of lexical premises (instantiated by parse)
• f is (LL atom corresponding to) sentential constituent
• M is meaning term produced by derivation
(Semantic) Ambiguity

Often (many) alternative derivations Γ Mi : f
each producing a dierent meaning term Mi for f
Need to nd all alternative derivations (eciently!)



Alternative Derivations: Modier Scope

Consider phrase alleged criminal from London
λx. criminal(x) : f
λP. alleged(P ) : f −◦ f
λP λx. from(l, x) ∧ P (x) : f −◦ f
Two derivations, resulting in:

1. λx. from(l, x) ∧ alleged(criminal)(x) : f
2. alleged(λx. from(l, x) ∧ criminal(x)) : f



Independence of Glue and Meaning

Original Glue (93) Curry-Howard Glue (97)
mixes meanings glue separates meanings glue
gjohn john : g
hf red f red : h
∀y. hy −◦ ∀x. (gx −◦ f see(x, y)) λyλx. see(x, y) : h −◦ (g −◦ f )

Some meaning separation rules:
• [[∀M. (rM −◦ ϕ)]]m = λM. [[ϕ]]m
• [[rM]]m = M
Some expressions can't be separated: gjohn −◦ f sleep(john)
Avoid these: derivations dependent on meanings,

and higher order unication needed to match meanings

Curry-Howard: good for understanding derivations
Original: good for understanding premises



Skeletons and modiers?



Glue Sales Pitch

Linguistically powerful exible approach
Interesting analyses of scope, control (Asudeh), event-based semantics
(Fry), intensional verbs (Dalrymple), context dependence,
coordination.
But many other phenomena still to do

Grammar semantics engineering
Applicable to grammars besides LFG based ones
Steep learning curve for writing lexical entries
But turns out to allow plentiful re-use of lingware

Can be implemented eciently



Conclusions

We missed Ulrik but managed to muddle through some basic denitions of
category theory.
Covered categories, functors, natural transformations and some examples.
Also discussed adjunctions and two of their easier examples: free monoids
and cartesian closedness.
We discussed an extended version of CurryHoward to relates logics, not
only to type theory, but also to categories and structures inside those
categories. Here's a graphic reminder of the main point.

As an novel and kind of unique application we mentione Glue Semantics
very briey.


Category Theory for All (NASSLLI 2012)

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