Kan Extensions of Institutions

Journal of Universal Computer Science, vol. 5, no. 8 (1999), 482-492

submitted: 7/8/99, accepted: 14/8/99, appeared: 28/8/99  Springer Pub. Co.

Grigore Rosu

Department of Computer Science & Engineering,

University of California at San Diego

[email protected]

Abstract: Institutions were intro duced by Goguen and Burstall [GB84 , GB85 , GB86 ,

GB92 ] to formally capture the notion of logical system. Interpreting institutions as func-

tors, and morphisms and representations of institutions as natural transformations, we

give elegant pro ofs for the completeness of the categories of institutions with morphisms

and representations, resp ectively, show that the dualitybetween morphisms and rep-

resentations of institutions comes from an adjointness b etween categories of functors,

and prove the co completeness of the categories of institutions over small signatures

with morphisms and representations, resp ectively.

Category: F.3, F.4

1 Intro duction

There are di erent logical systems successfully used in theoretical computer sci-

ence, such as rst and higher order logic, equational logic, Horn clause logic, tem-

p oral logics, mo dal logics, in nitary logics, and many others. As a consequence of

the fact that many general results of these logics are not dep endent on the partic-

ular ingredients of their underlying logic, abstracting Tarski's classic semantic

de nition of truth [Tar44], Goguen and Burstall [GB84, GB85, GB86, GB92]

develop ed the notion of institution to formalize the informal notion of \logical

system". The main requirement is the existence of a satisfaction relation b etween

mo dels and sentences which is consistent under change of notation.

Much interest has b een shown in the study of institutions since they rst

app eared in 1986. Institutions have b een given for lamb da calculus, higher order

logic with p olymorphic typ es, second order and mo dal logics. Mosses [Mos89]

shows that his uni ed algebras form an institution, Goguen [Gog91] shows that

his hidden-sorted equational logic is an institution, Mossakowski [Mos96] gives

hierarchies of institutions for total, partial and order-sorted logics, Rosu [Ros94]

gives an institution for order-sorted equational logic. Diaconescu, Goguen and

Stefaneas [DGS93] and Rosu [Ros99] use institutions to study mo dularization.

Diaconescu [Dia98] intro duces extra theory morphisms for institutions to give

logical semantics for multiparadigm languages like CafeOBJ [DF98]. Fiadeiro

and Sernadas [FS88] intro duces the notion of -institution based on deduction

rather than satisfaction, and Pawlowski [Paw96]intro duces the notion of context

institution to deal with variable contexts and substitutions. Cerioli and Meseguer

[CM97], Cerioli [Cer93], Tarlecki [Tar96a], Mossakowski [Mos96] study relation-

ships and translations b etween institutions. Muchinteresting work using institu-

tions has b een done byTarlecki [Tar84,Tar86a,Tar86b,Tar86c,Tar87,Tar96a]

and by Sannella and Tarlecki [ST86, ST87, ST88].

On leave from Fundamentals of Computing, Faculty of Mathematics, Universityof

Bucharest, Romania.

Rosu G.: Kan Extensions of Institutions 483

As suggested by Goguen and Burstall [GB92], an institution can b e regarded

as a functor from its category of signatures to some sp ecial category. This more

categorical view allows us to show that some known results on institutions are

instances of results in category theory, and also to obtain new results.

Two main apparently distinct maps b etween institutions are b eing considered

in the literature: institution morphisms due to Goguen and Burstall [GB92]

and institution representations due to Tarlecki [Tar87, Tar96a]. We show that

the categories of institutions with morphisms IN S and of institutions with

r epr

representations IN S , resp ectively, are sp ecial attened indexed categories,

thus their completeness following immediately.

Arrais and Fiadeiro [MJ96] showed that given an adjunction between two

categories of signatures, an institution morphism gives birth to an institution

representation and the vice-versa. We show that this duality is actually a natural

consequence of the fact that the adjointness b etween the categories of signatures

can b e contravariantly lifted to functor categories.

Given a functor between two categories of signatures, any institution over

the source signature category can b e extended to an institution over the target

signature category along that functor in two canonical ways given by the left and

the right Kan extensions, resp ectively. As a consequence, the categories of insti-

tutions over small signatures with morphisms and representations, resp ectively,

are co complete.

2 Category Theory

We assume the reader familiar with many categorical concepts. We use semicolon

for morphisms comp osition and it is written in diagrammatic order, that is, if

f : A ! B and g : B ! C are two morphisms then f ; g : A ! C is their

comp osition. We also use \;" for vertical comp osition of natural transformations

and \" for horizontal comp osition of natural transformations.

It is known that Cat and implicitly Cat and Set are b oth complete and

co complete. The reader is assumed familiar with limits and colimits in Set and

Cat .

2.1 Indexed Categories

Let Ind be any category, called \of indexes".

De nition 1. An indexed category is a functor C : Ind ! Cat. FlatC

is the category having pairs i; a as ob jects, where i is an ob ject in Ind and

0 0

a is an ob ject in C , and pairs ; f : i; a ! i ;a as morphisms, where

0 0

2 Indi; i and f 2 C a; C a .

The following two theorems show conditions under which the attened cat-

egory of an indexed category is complete or co complete [TBG91]:

Theorem 2. If C : Ind ! Cat is an indexedcategory such that Ind is com-

plete, C is complete for al l indices i 2jIndj, and C : C ! C is continuous

i j i

for al l index morphisms : i ! j , then FlatC is complete.

484 Rosu G.: Kan Extensions of Institutions

Theorem 3. If C : Ind ! Cat is an indexed category such that Ind is co-

complete, C is cocomplete for al l indices i 2jIndj, and C : C ! C has a left

i j i

adjoint for al l index morphisms : i ! j , then Flat C is cocomplete.

Given an indexed category C : Ind ! Cat , one can easily build another

op op

op op op

indexed category C : Ind ! Cat such that C is C and C : C !

op 0

C is C for every 2 Indi; i . The following corollaries are immediate

from Theorems 2 and 3, resp ectively:

Corollary 4. If C : Ind ! Cat is an indexedcategory such that Ind is com-

plete, C is cocomplete for al l indices i 2jIndj, and C : C ! C is cocontinuous

i j i

for al l index morphisms : i ! j , then FlatC is complete.

Corollary 5. If C : Ind ! Cat is an indexed category such that Ind is co-

complete, C is complete for al l indices i 2jIndj, and C : C ! C has a right

i j i

adjoint for al l index morphisms : i ! j , then Flat C is cocomplete.

2.2 Functor Categories and Kan Extensions

Let T be a category. For any category S, let T be the category of functors

from S to T and natural transformations, and for any functor : S ! S ,

S S 0 0

let T : T ! T be the functor de ned as T I = ; I for functors

0 0 0 0

I : S ! T and T = 1 for natural transformations : I I .

1 2

Also, let T : Cat ! Cat b e the functor that takes a category S to T and

: Cat ! Cat is an indexed a functor : S ! S to T . Then obviously T

category, and

Prop osition 6. If T is complete cocomplete then T is complete cocomplete

for any category S and T is continuous cocontinuous for any functor : S !

S .

Hint: The limits colimits in T are built \pointwise" see[Lan71 ], pg. 112.

De nition 7. Given functors K : S ! S and I : S ! T, a right Kan

1 2 1 1

extension of I along K is a pair containing a functor I : S ! T and

1 2 2

a natural transformation : K ; I I which is universal from T to I ,

2 1 1

0 0 0

that is, for every I : S ! T and : K ; I I there is a unique natural

2 1

2 2

0 0

transformation : I I such that = 1 ; . Dually, a left Kan

2 K

extension of I along K is a functor I : S ! T and a natural transformation

1 2 2

K 0

: I K ; I which is universal from I to T , that is, for every I : S ! T

1 2 1 2

0 0 0

and : I K ; I there is a unique natural transformation : I I such

1 2

2 2

that = ;1 .

The following ma jor result see [Lan71] plays an imp ortant role in our pap er:

Prop osition 8. Given a smal l category S ,

{ If T is complete then any functor I : S ! T has a right Kan extension

1 1

along any K : S ! S , and T has a right adjoint, and

1 2

{ If T is cocomplete then any functor I : S ! T has a left Kan extension

1 1

along any K : S ! S , and T has a left adjoint.

1 2

Rosu G.: Kan Extensions of Institutions 485

The following theorem gives other conditions see Corollary 11 under which

Kan extensions exist.

Theorem 9. T contravariantly lifts adjoints to functor categories.

Hint: If h; ; ; i : S ! S is an adjointness then so is hT ; T ; T ; T i,from

S S

0 0

T ! T , where T = 1 and T = 1 for every functors I : S ! T

I I I I

0 0

and I : S ! T.

Then within the same notations,

0 0

Corollary 10. Nat ; I; I ' NatI; ; I and this bijection is natural in I

and I .

More precisely, a natural transformation : ; I I is taken to 1 ; 1

and conversely, a natural transformation : I ; I is taken to 1 ;

. 1

Corollary 11. Given any functor I : S ! T, then

1 1

{ I has a right Kan extension along any K : S ! S which has a left adjoint,

1 1 2

and T has a right adjoint,

{ I has a left Kan extension along any K : S ! S which has a right adjoint,

1 1 2

and T has a left adjoint.

2.3 Twisted Relations

Twisted relations are intro duced in [GB92]. We show that they give a complete

and co complete category.

De nition 12. Let Trel b e the category of \twisted relations", having triples

hA; R;Bi as ob jects, where A is a category, B is a set and R jAjB , and pairs

0 0 0 0

hF; g i : hA; R;Bi!hA ; R ;B i as morphisms, where F : A ! A is a functor

and g : B ! B is a function such that the diagram

jAj

O B

g F

jA j

commutes.

Let Lef t : Trel ! Cat take a triple hA; R;Bi to A and a morphism hF; g i

to F , and let Rig ht : Trel ! Set take a triple hA; R;Bi to B and a morphism

hF; g i to g . It is straightforward that Lef t and Rig ht are functors.

Prop osition 13. Trel is both complete and cocomplete.

486 Rosu G.: Kan Extensions of Institutions

Proof. Let J b e a small category and D : J ! Trel b e a functor.

First, let A; fF g and B; fg g b e limits of D ; Lef t and D ; Rig ht,

j j

j 2jJj j 2jJj

resp ectively they exist b ecause Cat and Set are complete, and let R

jAjB b e the relation de ned as a; b 2R i a ;g b 2R for each j 2jJj

j j j

and each a 2jA j with F a =a. Notice that for every a 2jAj there are some

j j j j

j 2jJj and a 2jA j such that F a =a b ecause of the way in which colimits

j j j j

are built in Cat. It is easy now to check that hA; R;Bi; fhF ;g ig is a

j j

j 2jJj

limit of D . Therefore, Trel is complete.

Let fF g ;A and fg g ;B be colimits of D ; Lef t and D ; Rig ht,

j j

j 2jJj j 2jJj

resp ectively they exist b ecause Cat and Set are co complete, and let R

jAjB be the relation de ned as a; b 2 R i F a ;b 2 R for each

j j j j

j 2 jJj and each b 2 B with g b = b. As b efore, notice that for every

j j j j

b 2 B there are some j 2 jJj and b 2 B such that g b = b, and that

j j j j

fhF ;g ig ; hA; R;Bi is a colimit of D .Thus Trel is co complete.

j j

j 2jJj

3 Institutions

Institutions were intro duced by Goguen and Burstall [GB92] to formally capture

the notion of logical system.

De nition 14. An institution Sign; Mo d; Sen; j= consists of a category

Sign whose ob jects are called signatures, a functor Mo d : Sign ! Cat

giving for each signature a category of -mo dels, a functor Sen : Sign !

Set giving for each signature a set of -sentences, and a -indexed relation

j= = fj= j 2 Signg, where j= jMo d j Sen , such that for each

signature morphism ' : ! , the following diagram commutes,

jMo d j Sen

Mo d' Sen'

0 0

jMo d j Sen

that is, the following Satisfaction Condition

0 0

m j= Sen'a i Mo d'm j= a

0 0

holds for each m 2jMo d j and each e 2 Sen .

Prop osition 15. There is a bijection between institutions over the signature

Sign and functors Sign ! Trel.

Proof. Given an institution Sign; Mo d; Sen; j=, then one can build a func-

tor I : Sign ! Trel that takes every signature 2 jSignj to the triple

hMo d ; j= ; Sen i, and every morphism of signatures ' : ! to the

\twisted" morphism hMo d'; Sen 'i.

Conversely, given a functor I : Sign ! Trel, one can build an institu-

tion Sign; Mo d; Sen; j= such that for every 2 jSignj, Mo d = A ,

Actually we mean one-to-one corresp ondence b etween classes.

Rosu G.: Kan Extensions of Institutions 487

Sen =B , and j= = R , where I =hA ; R ;B i, and such that for

every morphism of signatures ' : ! , Mo d' = F and Sen' = g ,

' '

where I '=hF ;g i.

' '

Because of this bijection, we can interchangeably use any of the tuple or functor

notation when referring to institutions.

Some other more categorical views of institutions are explored in [GB92],

where the target category is a comma category. Dep ending on the functors in-

volved in the comma category, one can obtain institutions as in our approach,

institutions with no morphisms b etween mo dels, or even institutions with mor-

phisms b etween sentences. The morphisms b etween sentences are thoughtofas

\pro ofs", as advo cated by Lamb ek and Phil Scott [LS86].

3.1 Institution Morphisms

Institution morphisms were intro duced together with the institutions in [GB92].

De nition 16. Given two institutions I = Sign; Mo d; Sen; j= and I =

0 0 0

0 0

Sign ; Mo d ; Sen ; j= , an institution morphism from I to I consists of a

0 0

functor : Sign ! Sign, a natural transformation : Mo d ; Mo d, and a

natural transformation : ; Sen Sen , such that the following Satisfaction

0 0

0 0 0 0

Condition holds for each 2jSign j, m 2jMo d j, and e 2 Sen :

0 0 0

m j= e i m j= e

Let IN S denote the category of institutions and institution morphisms.

0 0

Intuitively, a morphism from I to I is a pro jection of the logic I into the

logic I .

Theorem 17. IN S is isomorphic to FlatTrel .

Proof. By Prop osition 15, any institution Sign; Mo d; Sen; j= is taken bijec-

tively to a pair Sign; I : Sign ! Trel, that is, an ob ject of FlatTrel . If

0 0

h ; ; i is a morphism from Sign ; Mo d ; Sen ; j= toSign; Mo d; Sen; j=,

op 0

then let h ; i b e a morphism in FlatTrel , from Sign ; I toSign; I

SignO L LLL

LI LLL

LL&

rel r9T rrr rrr rr 0

r I

Sign

0 0 0

where : ; I I is the natural transformation de ned as = h ; i.

It is easy but meticulous to show that is a natural transformation and that

this map indeed gives an isomorphism b etween IN S and FlatTrel .

488 Rosu G.: Kan Extensions of Institutions

Therefore, we can use morphisms in FlatTrel instead of institution mor-

phisms whenever suchchoice simpli es the exp osition. The following result was

proved for the rst time byTarlecki in [Tar86a] see also [Tar96a,Tar96b]:

Corollary 18. IN S is complete.

Proof. By Prop osition 13, Prop osition 6 and Corollary 4, one obtains that the

category FlatTrel is complete, so IN S is complete.

3.2 Institution Representations

A slightly di erent notion of mapping b etween institutions, namely institution

representation, was intro duced byTarlecki in [Tar87] see also [Tar96a,Tar96b].

This is a sp ecial case of Meseguer's map of institutions [Mes89]:

De nition 19. Given two institutions I = Sign; Mo d; Sen; j= and I =

0 0

Sign ; Mo d ; Sen ; j= , an institution representation from I to I consists

0 0

of : Sign ! Sign , a natural transformation : ; Mo d Mo d, and a nat-

ural transformation : Sen ; Sen , such that the following Representation

0 0

Condition holds for each 2jSignj, m 2jMo d j, and e 2 Sen :

0 0

m j= e i m j= e

r epr

Let IN S denote the category of institutions and institution representations.

0 0

Intuitively, a representation from I to I is an enco ding of the logic I into

the logic I .

r epr

Theorem 20. IN S is isomorphic to FlatTrel .

Proof. Similarly but dually to the pro of of Theorem 17, if h; ; i is a repre-

0 0

sentation from Sign; Mo d; Sen; j= to Sign ; Mo d ; Sen ; j= , then let h; i

b e a morphism in FlatTrel , from Sign; I toSign ; I :

Sign L LLL

LI LLL

LL&

rel r9T rrr rrr rr 0

r I

Sign

such that : I; I is the natural transformation de ned as = h ; i.

r epr

This de nes an isomorphism b etween IN S and FlatTrel . The rest of the

pro of is left to the reader.

Thus, we can use morphisms in FlatTrel instead of the more complicated

institution representations whenever suchchoice simpli es the exp osition. Com-

r epr

pleteness of IN S was shown for the rst time byTarlecki in [Tar96b]:

r epr

Corollary 21. IN S is complete.

Proof. By Prop osition 13, Prop osition 6, Theorem 2 and the fact that Cat is

r epr

complete, FlatTrel is complete, so IN S is complete.

Rosu G.: Kan Extensions of Institutions 489

3.3 Duality Between Morphisms and Representations

Arrais and Fiadeiro [MJ96] observed for the rst time that given an adjoint

pair of functors b etween two categories of signatures, there is a dualitybetween

the institution morphisms and the institution representations asso ciated. We are

pleased to notice that this nice result is a natural consequence of the fact that the

functor Trel contravariantly lifts adjoint pairs to functor categories Theorem

0 0

Theorem 22. If : Sign ! Sign is a left adjoint to : Sign ! Sign then

for any institutions I : Sign ! Trel and I : Sign ! Trel there is a bijection

between institution morphisms h ; i : I ! I and institution representations

0 0

h; i : I ! I . Moreover, this bijection is natural in I and I .

Proof. It follows easily by Corollary 10, as the institution morphisms and the

institution representations are ordinary morphisms in FlatTrel Theorem

Theorem 20, resp ectively. 17 and Flat Trel

The bijection in Corollary 10 takes a natural transformation : ; I

I to 1 ; 1 , and its inverse takes a natural transformation : I

, where and are the unit and the counit of the ; I to 1 ; 1

adjunction, resp ectively. Translating that in a more institutional language, by

the construction of isomorphisms in Theorems 17 and 20, one gets that:

0 0 0

1. institution morphism h ; ; i : I I yields an institution representation

0 0 0

; Mo d and = h; ; i : I ! I , where : Sen ;

for all 2jSignj, and

2. institution representation h; ; i : I I yields an institution morphism

0 0 0 0 0

h ; ; i : I ! I , where for all 2jSign j, = ; Sen and

= Mo d ; .

This is exactly the construction describ ed in [MJ96].

4 Kan Extensions of Institutions

Provided a signature translation, a whole institution can be translated in two

distinct but canonical ways, given by the two Kan extensions asso ciated.

Prop osition 23. Given a smal l category Sign and a functor K : Sign ! Sign ,

any institution I : Sign ! Trel has both a right and a left Kan extension along

K , and the functor Trel has both a right and a left adjoint.

Proof. It follows from Prop osition 8, noticing that Trel is b oth complete and

co complete Prop osition 13.

De nition 24. Let SCat b e the category of small categories, SIN S b e the cat-

egory of institutions over small signature categories and institution morphisms,

r epr

and SIN S be the category of institutions over small signature categories

and institution representations.

490 Rosu G.: Kan Extensions of Institutions

It is easy to see that SCat is b oth complete and co complete, and that The-

orems 17 and 20 can b e adapted to categories of small signatures, thus getting

r epr

that SIN S and SIN S are b oth complete. It is not known if IN S and/or

r epr

IN S are/is co complete. The following result seems to b e new:

r epr

Theorem 25. SIN S and SIN S arecocomplete.

Proof. By Theorems 17 and 20, it suces to show that Flat Trel and

FlatTrel are co complete. This follows from Corollary 5 and Theorem 3, notic-

Sign

ing that SCat is co complete, that Trel is b oth co complete and complete

Prop osition 6 for T = Trel, using Prop osition 13 for all signature categories

Sign, and that Trel has a left adjoint and a right adjoint Prop osition 23.

5 Conclusion

This pap er can b e regarded as an application of category theory to the theory of

institutions. Organizing institutions as functors b etween certain categories, we

b oth gave elegant categorical pro ofs to known facts such as the completeness of

the category of institutions, the dualitybetween morphisms and representations

of institutions and obtained new conjectured results the co completeness of

the category of institutions. But from the author's p oint of view, the most

fascinating result is Prop osition 23, which philosophically says that provided a

logic and a translation of its syntax to another syntax, then the whole logic can

b e translated to a logic over the new syntax in two cannonical ways.

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