
Key Questions for this tutorial
• What is Category Theory?

• Why should we be interested in Category Theory?

• How much Category Theory is it useful to know?

• What kinds of things can you do with Category Theory in
Software Engineering?

• (for the ASE audience)
Does Category Theory help us to automate things?



By way of introduction...
• An explanation of “Colimits”

A B

The colimit f
of A and B A co-cone
over A and B

• My frustration:
 Reading a maths books (especially category theory books!) is like reading a program
without any of the supporting documentation. There’s lots of deﬁnitions, lemmas, proofs,
and so on, but no indication of what it’s all for, or why it’s written the way it is.
 This also applies to many software engineering papers that explore formal foundations.


Outline
(1) An introduction to categories you
 Definitions are
 Some simple examples here

(2) Motivations
 Why is category theory so useful in mathematics?
 Why is category theory relevant to software engineering?

(3) Enough category theory to get by
 some important universal mapping properties
 constructiveness and completeness

(4) Applying category theory to specifications
 Specification morphisms
 Modular Specifications
 Tools based on category theory


Definition of a Category
• A category consists of: A f B
 a class of objects
 a class of morphisms (“arrows”)
 for each morphism, f, one object as the domain of f A
and one object as the codomain of f.
 for each object, A, an identity morphism which has IDA
domain A and codomain A. (“IDA”)
 for each pair of morphisms f:A→B and g:B→C, (i.e. f g
cod(f)=dom(g)), a composite morphism, g  f: A→C
gf

• With these rules:
 Identity composition: For each morphism f:A→B, hg
f  IDA = f and IDB  f = f f g h
 Associativity: For each set of morphisms f:A→B, g:B
→C, h:C→D, gf
(h  g)  f = h  (g  f) (h  g)  f = h  (g  f)


Understanding the definition
Which of these can be valid categories?

Note: In this notation, the identity morphisms are assumed.


Understanding the definition
Proof that is not a category:

Composition: Associativity:
g f  h = IDB h  f  g = (h  f)  g
IDA IDB
f  g = IDB = IDA  g
f
h  f = IDA =g
A B
g  f = IDA h  f  g = h  (f  g)


h = h  IDB
=h

okay so far
Hence: g = h

not okay
Note : h" f = g" f # h = g, although it may in some categories.
/

Hence, can be a category.

! © Steve Easterbrook, 1999 7


Challenge Question
(For the experts only)

Can this be a category?
These are
not
identities



Example category 1
• The category of sets (actually, “functions on sets”)
 objects are sets
 morphisms are functions between sets

E.g. E.g.
“best friend” Temperatures
alice kurt
sam bob
eric
stan
What are
earl
measure
in ºF measure the missing
in ºC morphisms?

“what best “likes for
friend likes for breakfast” convert round
breakfast” ºF to ºC
tea coffee cast to real
juice convert Integers
Real
water ºC to ºF numbers



Example category 2
• Any partial order (P, ≤)
 Objects are the elements of the partial order
 Morphisms represent the ≤ relation.
 Composition works because of the transitivity of ≤

E.g.
The partial order n, formed from the ﬁrst n natural numbers
Here, n = 4

1

2
0 3

4



Outline
(1) An introduction to categories
 Deﬁnitions
 Some simple examples

(2) Motivations you
 Why is category theory so useful in mathematics? are
 Why is category theory relevant to software engineering? here




So what? (for the mathematician)
• Category theory is a convenient new language
 It puts existing mathematical results into perspective
 It gives an appreciation of the unity of modern mathematics

• Reasons to study it
 As a language, it offers economy of thought and expression
 It reveals common ideas in (ostensibly) unrelated areas of mathematics
 A single result proved in category theory generates many results in different areas of
mathematics
 Duality: for every categorical construct, there is a dual, formed by reversing all the
morphisms.
 Difﬁcult 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’

“To each species of mathematical structure, there corresponds a category, whose
objects have that structure, and whose morphisms preserve it” - Goguen


Some more definitions
• Discrete category: Example:
 All the morphisms are identities IDA IDB

IDC

• Connected category:
 For every pair of objects, there is at least
one morphism between them
Example:
original category: full sub-category:

• Full sub-category:
 A selection of objects from a category,
functor
together with all the morphisms between
them.



Inverses and Isomorphisms
• Identity morphism:
 For each object X, there is an identity morphism, IDX, such that:
f
 if f is a morphism with domain X, f  IDX = f X
 if g is a morphism with codomain X, IDX  g = g g
IDX
• Inverse
 g:B→A is an inverse for f:A→B if:
f  g = IDB IDA IDB
A f B
g  f = IDA
 If it exists, the inverse of f is denoted f-1 g
 A morphism can have at most one inverse

• Isomorphism
 If f has an inverse, then it is said to be an isomorphism
 If f:A→B is an isomorphism, then A and B are said to be isomorphic



Example category 3
• Category of geometric shapes (Euclid’s category)
 objects are polygonal ﬁgures drawn on a plane
 morphisms are geometric translations of all the points on the polygon such that distances
between points are preserved.
 Objects that are isomorphic in this category are called ‘congruent ﬁgures’

p

q'
q r'
P'
r

p

q'
q r'
P'
r



Example category 4
• Category of algebras
 Each object is a sort, with a binary function over that sort
 Each morphism is a translation from one algebra to another, preserving the structure

Works because
E.g. e(a+b) = ea x eb
E.g.

exponentiation
(!,+) ({odd, even},+)
(" >0 ,!)
doubling

(!,+)
({pos, neg},!)
Works because
2(a+b) = 2a + 2b



Functors
• Definition of functor:
 Consider the category in which the objects are categories and the morphisms are mappings
between categories. The morphisms in such a category are known as functors.
 Given two categories, C and D, a functor F:C→D maps each morphism of C onto a
morphism of D, such that:
F preserves identities - i.e. if x is a C-identity, then F(x) is a D-identity
F preserves composition - i.e F(f  g) = F(f)  F(g)

• Example functor
 From the category of topological spaces and continuous maps
to the category of sets of points and functions between them


So what? (for the software engineer)
• Category theory is ideal for:
 Reasoning about structure and the mappings that preserve structure
 Abstracting away from details.
 Automation (constructive methods exists for many useful categorical structures)

• Applications of Category theory in software engineering
 The category of algebraic specifications - category theory can be used to represent
composition and refinement
 The category of temporal logic specifications - category theory can be used to build
modular specifications and decompose system properties across them
 Automata theory - category theory offers a new way of comparing automata
 Logic as a category - can represent a logical system as a category, and construct proofs
using universal constructs in category theory (“diagram chasing”).
 The category of logics - theorem provers in different logic systems can be hooked together
through ‘institution morphisms’
 Functional Programming - type theory, programming language semantics, etc


Modularity in Software Engineering
• Reasons for wanting modularization
 Splitting the workload into workpieces
“decompose the process”
 Splitting the system into system pieces (components) Most of the category
theory work has
“decompose the implementation”
addressed this one
 Splitting the problem domain into separate concerns
“decompose the requirements”

• Resulting benefits
 Information hiding
 Compositional veriﬁcation
 Compositional reﬁnement

• Generalizable approaches:
 Semi-formal - Viewpoints framework
 Formal - Category Theory


Building blocks
• Need to express:
 Modules (Interface + Structure + Behavior)
 Module Interconnections
 Operations on modules (e.g. compose two modules to form a third)
interface

interface
structure structure
interconnection
behavior behavior

new
module


Outline
 Deﬁnitions

(2) Motivations

(3) Enough category theory to get by you
 some important universal mapping properties are
 constructiveness and completeness here



Enough Category Theory to get by...
• Universal Constructs
 General properties that apply to all objects in a category
 Each construct has a dual, formed by reversing the morphisms
 Examples:
– initial and terminal objects
These are the building
– pushouts and pullbacks blocks for manipulating
– colimits and limits specification
– co-completeness and completeness structures

• Higher order constructs
 Can form a category of categories. The morphisms in this category are called functors.
 Can form a category of functors. The morphisms in this category are called natural
transformations.
 Can consider inverses of functors (and hence isomorphic categories). Usually, a weaker
notion than isomorphism is used, namely adjoint functors.



Initial and Terminal Objects
• Initial objects • Terminal objects
An object S is said to be initial if for every other An object T is said to be terminal if for every
object X there is exactly one morphism f:S→ other object X there is exactly one
X morphism f:X→T

• Examples • Example
 The number 0 in this category:  Any singleton set in the category of sets

1
2
0 3
 The empty set {} in the category of sets

Uniqueness (up to isomorphism):
 If T1 and T2 are both terminal objects, then there is exactly one
morphism between them, and it is an isomorphism T2 IDT2
IDT1 T1 f
 Why? Because there is exactly one morphism each of f:T1→T2,
g:T2→T1, h:T1→T1, and j:T2→T2, where h and j are identities. g
 Same applies to initial objects



Pushouts and Pullbacks
• Pushout • Pullback
The pushout for two morphisms f:A→B and g:A The pullback for two morphisms f:A→C and g:B
→C is an object D, and two morphisms d1:B →C is an object D, and two morphisms d1:D
→D and d2:C→D, such that the square →A and d2:D→B, such that the square
commutes… commutes…

A f B D’ d1’
d’
d1 A
g d1 d2’
d1’ D
D d2 f
d2
C d’
g
d2’ B C
D’
… and D is the initial object in the full … and D is the terminal object in the full
subcategory of all such candidates D’ subcategory of all such candidates D’
(I.e. for all objects D’ with morphisms d1’ and
d2’, there is a unique morphism from D to
D’)


Products and Coproducts
• Coproduct • Product
The coproduct of a family of objects Ai is an The product of a family of objects Ai is an object
object P and a set of morphisms gi:Ai→P P and a set of morphisms gi:P→ Ai

A1 A1

g1 P g1 P
P’ P’
g2
g2

A2 A2
… and P is the initial object in the full … and P is the terminal object in the full
subcategory of all such candidates P’ subcategory of all such candidates P’

• Coproduct vs. Pushout • Product vs. Pullback
 Pushout is a universal property of any two  Pullback is a universal property of any two
morphisms with a common domain morphisms with a common codomain
 Coproduct is a universal property of any set  Product is a universal property of any set of
of objects objects


Example products
• In the category of sets: • In the category of
 constructed as the cartesian product geometric spaces:

• In the category of logical
propositions: p'

a∧b b a∧b b In any given category,
some products might not
exist. It is useful to know
a a whether they all do.


Example co-product & pushout
{a,b,c} {x,y}
• Coproducts on the category of
sets:
 Constructed by taking the disjoint sum
{a,b,c,x,y}

A B
k
x b
s
• Pushouts on the category of y e

sets:
 Union of:
 Pairs of elements from B and C that are the
images of the same element in A C P
 Plus all the remaining elements of B and C a bB
s kB (sB, sC)
e (eB, eC) a
C



Limits and Colimits
• Colimits • Limits
 initial objects, pushouts and coproducts are  terminal objects, pullbacks and products are
all special cases of colimits. all special cases of limits.
 Colimits are deﬁned over any diagram  Limits are deﬁned over any diagram

For any diagram containing objects Ai and For any diagram containing objects Ai and
morphisms ai, the colimit of this diagram is morphisms ai, the limit of this diagram is an
an object L and a family of morphisms li, object L and a family of morphisms li, such
such that for each li: Ai→L, lj: Aj→L, and that for each li:L→Ai, lj:L→Aj, and ax:Ai→
ax:Ai→Aj, then lj  ax = li Aj, then ax  li = lj

Ai Ai

ax li L ax li L
L’ L’
lj lj

Aj Aj
… and L is the initial object in the full … and L is the terminal object in the full
subcategory of all such candidates L’ subcategory of all such candidates L’


Completeness and Co-completeness
• It is useful to know for a given category which universal
constructs exist:
 If a category has a terminal object and all pullbacks exist, then all finite limits exist
– Hence it is finitely complete
 If a category has an initial object and all pushouts exist, then all finite colimits exist
– Hence it is finitely cocomplete

• Proofs are usually constructive
 I.e. give a method for computing all pullbacks (pushouts)
 The constructive proof is the basis for automated generation of limits (colimits)

• Obvious application
 If your objects are specifications, then:
– colimits are the integration of specifications
– limits are the overlaps between specifications



Outline
 Deﬁnitions

(2) Motivations


you
are
here


(Recall...) Algebraic Specifications
• A signature is a pair <S, Ω>
where S is a set of sorts, and Ω is a set of operations over those sorts
• A specification is a pair <Σ, Φ>
describes algebras over the signature Σ that satisfy the axioms Φ

Spec Container
Signature sort Elem, Cont
op empty: Cont
op single: Elem -> Cont
op merge: Cont, Cont -> Cont
Body axiom merge(empty, e) = e
axiom merge(e, empty) = e
end-spec
• Semantically:
 We are modeling programs as algebras
 A speciﬁcation deﬁnes a class of algebras (programs)



Specification morphisms
• Specfication morphisms
 Consider the category in which the objects are specifications
 The morphisms translate the vocabulary of one specification into the vocabulary of
another, preserving the (truth of the) axioms

• Actually, there are two parts:
 Signature morphism: a vocabulary mapping
– maps the sorts and operations from one spec to another
– must preserve the rank of each operation
 Specification morphism: a signature morphism for which each axiom of the first
specification maps to a theorem of the second specification

• Proof obligations
 There will be a bunch of proof obligations with each morphism, because of the need to
check the axioms have been translated into theorems
 A theorem prover comes in handy here.


Example
Spec Container
sort Elem, Cont
op empty: Cont These comprise the
op single: Elem -> Cont signature morphism
(Note each spec has it’s
axiom merge(empty, e) = e
axiom merge(e, empty) = e own namespace)
end-spec

Spec List
sort Elem, List
op null: List
op single: Elem -> List
op append: List, List -> List
op head: List -> Elem
op tail: List -> List
axiom head(single(e)) = e
These axioms must axiom tail(single(e)) = null
be true down here axiom append(single(head(l)), tail(l)) = l
end-spec
(after translation)


What do we gain?
• Three simple horizontal composition primitives:
 Translate: an isomorphic copy (just a renaming)
– can test whether two specifications are equivalent
 Import: include one specification in another (with renaming)
– for extending specifications with additional services
 Union (colimit): Compose two specifications to make a larger one
– system integration

• One simple vertical composition primitive:
 refinement: mapping between a specification and its implementation
– introduce detail, make design choices, add constraints, etc.
– (may want to use different languages, e.g. refinement is a program)



Example colimit (pushout)
Spec List
Spec Container sort Elem, List
sort A, B op null: List
op x: B op head: List -> Elem
end-spec op tail: List -> List
op cons: Elem, List -> List
axiom head(cons(e, l)) = e
axiom tail(cons(e, l)) = l
axiom cons(head(l), tail(l)) = l
axiom tail(cons(e, null)) = null
end-spec

Spec Container
sort Elem, Cont
op empty: Cont
op single: Elem -> Cont
axiom merge(empty, e) = e New spec is lists
axiom merge(e, empty) = e with two new operations,
end-spec “single” and “merge”


(Recall…) Temporal Logic Specs
• A signature is a pair <S, Ω>
where S is a set of sorts, and Ω is a set of operations over those sorts

• A specification is a 4-tuple <Σ, ATT, EV, AX>
Σ is the signature
These three comprise the
ATT is a set of attributes vocabulary of the specification
EV is a set of events
AX is a set of axioms expressed in temporal logic
Assume some usual temporal
logic operators, e.g.
• Semantically: always
eventually
 We are modeling programs as state machines
 A speciﬁcation describes a class of state machines that obey the axioms

• (A minor complication)
 Need to worry about locality of events



Expressing modules
• Want to generalize the notion of a module
 Explicitly declare interfaces, with constraints on imported and exported resources
 Hence the interface itself is a speciﬁcation (actually 2 specifications)
(Ehrig & Mahr use algebraic specs; Michel & Wiels use temporal logic specs)

Parameters of Resources provided
this module by this module
parameters export
(identifies
IN-OUT
parameters)

import body Realization of the
Resources to
resources declared
be provided to
in the export part
this module by
(may contain
others
hidden sorts and
operations)


Examples
• The approach works for many different kinds of module:
E.g. function modules E.g. data types
datalists; datalist data;
lists
equality; sorting equality;
list opns;
total order; function; ordering;

implement’n implement’n
lists;
of sorting same of list
list opns;
function operations

E.g. predicates E.g. state machines

SORTED: common output
data
list->bool events events

input state
list(data) same
events machine



Composing modules
E.g. import (“uses”): E.g. union (colimit):
par par1 exp1 par0 exp0

imp0 bod0
imp1 bod1

par2 exp2
par1 exp1 par2 exp2

imp2 bod2 bod
imp1 bod1 imp2 bod2

new module


Advanced Topics
• Logic engineering
 Language translation
– from one logic to another
– from one specification language to another
 Aim is to characterize logics as:
– signatures (alphabet of non-logical symbols)
– consequence relations
 Then an institution morphism allows you to translate from one logic to another whilst
preserving consequence

• Natural Transformations of refinements
 If a system specification is a category, and the relationship between the specification and its
refinement is a functor…
 …then the relationship between alternative refinements of the same specification is a
natural transformation.



(suggested)
Future Research Issues
• Compositional Verification in Practice
 E.g. How much does the choice of modularization affect it
 Which kinds of verification properties can be decomposed, and which cannot?
 How do we deal systemic properties (e.g. fairness)

• Evolving Specifications
 How do you represent and reason about (non-correctness preserving) change?
 How resilient is a modular specification to different kinds of change request

• Dealing with inconsistencies
 Specification morphisms only work if the specifications are consistent
 Can we weaken the “correct by construction” approach?



Summary
• Category Theory basis
 Simple definition: class of objects + class of arrows (morphisms)
 A category must obey identity, composition and associativity rules

• Category theory is useful in mathematics…
 Unifying language for talking about many different mathematical structures
 Provides precise definition for many abstract concepts (e.g. isomorphism)
 Framework for comparing mathematical structures

• Category theory is useful in software engineering
 Modeling and reasoning about structure
 Provides precise notions of modularity and composition
 Specification morphisms relate vocabulary and properties of specifications
 Constructive approach lends itself to automation



Answer to challenge question:

YES!
(proof left as an exercise for the audience*)


Introduction to Category Theory for software engineers