On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final...

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On the Greatest Fixed Point of a Finitary Set Functor James Worrell Oxford University Computing Laboratory, UK

Transcript of On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final...

Page 1: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

On the Greatest Fixed Point of a Finitary Set

Functor

James WorrellOxford University Computing Laboratory, UK

Page 2: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Greatest fixed points

A fixed point of an endofunctor

��� � � �

is a pair

���� � � � ��

,

where

is an object of

and

is an isomorphism. A homomorphism of

fixed points

����

and

�� � � is an arrow

� � � � such that

��

��

����� ��� �

��� ���� ��

Defn. The greatest fixed point of

is the final object in the above

category of fixed points.

Page 3: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Examples

Non-well-founded sets: The greatest fixed point of the powerset functor� � ���� � � � �� � � �

is a model of ZF

+ anti-foundation axiom.

Reactive processes: The class of

-labelled transition systems

quotiented by bisimilarity is a greatest fixed point of

� � �� �

.

Domain equations: Suppose

is algebraically compact. Let � ��� � � � �

, and define

� � �� � � � ��� � �

by

� �� � � � � � � � � ���� � ��

The greatest fixed point of

has the form

�� � �

where

� � � �� � �

.

Scott’s

� � model of the�

-calculus can be seen in this way.

Page 4: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Coalgebras

A coalgebra of an endofunctor generalises a post-fixed point of an

monotone map.

A coalgebra of an endofunctor

��� � � �

is a pair

� ��� � � � ��

,

where

, the carrier of the coalgebra, is an object of�

, and

, the

structure map, is an arrow of

. A homomorphism of

-coalgebras����

and

� � � � is an arrow

� � � � such that

��

��

����� ��� �

��� ���� ��

Thm. (Lambek) The final object in the category

�� of

-coalgebras is a

fixed point of

.

Page 5: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Finitary functors

A cardinal � is regular if it is not the sum of fewer than � strictly smaller

cardinals. For a regular cardinal � we say that a partially ordered set

is

�-directed if each subset of

of size strictly less than � has an upper

bound in

.

A functor

�� � � �

is �-accessible if it preserves colimits of those

diagrams indexed by �-directed posets. An �-accessible functor is

sometimes called finitary.

Page 6: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

GFP existence: a five-line proof

Prop. Let

� � ��� � � �� �

be an endofunctor. Then the forgetful functor

� � �� �� � �� �

creates (and preserves) colimits.

Thm. (Barr) If

� � �� � � �� �

is accessible then there is a final�

-coalgebra.

Pf. By the special adjoint functor theorem we must show that

�� �� is

cocomplete, well-copowered and has a set of generators. The

cocompleteness and well-copoweredness follow from the same facts for

�� �

by the above proposition. Makkai and Pare’s weighted bilimit

theorem implies that

�� �� is accessible and thus has a set of generators.

Page 7: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A more concrete approach

The final sequence of

is an ordinal-indexed sequence of objects

� � �� �

,

with maps

� �� � � �� � � �� � � � , uniquely defined by the following

conditions (where

� � �

):

� FS-1

� �� �� � � � � ��

;

� FS-2

�� �� � � � ��

;

� FS-3

�� � �� � �� ;

� FS-4 if

is a limit ordinal, then

� �� � � �� � � �� � � � is a limit

cone.

Prop. If

� � � is an isomorphism, then

� �� �� � � � �

is a final

coalgebra.

Page 8: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Motivating example

Our leading example of a finitary set functor is the finite powerset functor�� �� � � ��� �

. For a set

,

� �

is the collection of finite subsets of

.

For a function

� � � �

,

� � � � � � �is defined by� � �� � ��

.

A

-coalgebra

���� � can be seen as a graph or a transition system.

Next we investigate the final sequence on

.

Page 9: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Tree bisimulations

We consider trees as directed graphs with a distinguished root node from

which every other node is reachable by a unique path.

A relation

on the set of nodes of a tree is a tree bisimulation if � ���

implies the respective parents of � and � are related, each child of � is

-related to a child of � , and each child of � is� �

-related to a child of

�. We call a tree strongly extensional if no two distinct nodes are related

by a bisimulation. For any tree

the union of all tree bisimulations is an

equivalence, and the quotient of�

by this equivalence is strongly

extensional.

Remark. An infinite-depth tree can be extensional without being strongly

extensional.

Page 10: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Tree representation

Write

� � �

for the final object in

�� �

. For each � � � there is a bijection

between

�� �

and the set of finitely branching, strongly extensional trees

of depth no greater than �, whose depth- � nodes are labelled

. By

induction: if � � � � �� ��� � �� �

correspond to trees� � � � �� �� , then� � � � � �� �� � � �� � �

corresponds to

������

��

� � � � ��

The projection map

� � � takes a tree in

� � � �

, removes the

depth-

� � � �

nodes and computes the strongly extensional quotient.

Page 11: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Growing trees

Below we draw a row of trees taken respectively from

� � �� � �� � � �

and� � �

, with each tree projecting down to the tree to the left.

� � �

��

��

��

��

��

��

��

��

��

� � � � � �

� � ��

Page 12: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

In the limit

We can imagine the trees as projections of the following compactlybranching infinite-depth tree in

� � �

.

��

��

��

��

��

��

��

��

���������������������

� � � � � � �

� � �

� �� � �

Page 13: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A metric on trees

Given trees

�� � �

, write

� � � � �

if the restrictions of�

and

� �

to depth � have

the same strongly extensional quotient, and define a pseudo-metric

��� on

the class of strongly extensional trees by

�� � �� � � � ��� � � � � � � � � � ��

Page 14: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

To infinity and beyond!

� � �

is the set of compactly branching, strongly extensional trees of

possibly infinite depth.

� � � �

is the set of compactly branching strongly, strongly extensional

trees that are finitely branching at the root.

� � � � �

is the set of compactly branching strongly, strongly extensional

trees that are finitely branching to depth �.

� � � � �

is the set of finitely branching, strongly extensional trees.

Thus the final sequence of�

consists of � steps of building up higher and

higher trees, followed by � steps of pruning level by level.

Page 15: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

The general case

We show that the final sequence

� � ��� �� �

of a finitary functor� � �� � � �� �

stabilises in � � � � � steps.

Note that each

-coalgebra

� �� � extends to a cone

� � � � � � � ��

over the final sequence, where � � � � � � � � � �.

��

��

� ��� � �

� ���

� ���� � � �

� � ���

� � ���� � � �

� � ���

� � ���� � � �

� ���� � ���� � ���� � � ���

Page 16: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

The parsing map

Prop.

� � � is injective (and hence

�� is injective for all � � �).

Choose a parsing map � � � �� � � � � �� , where � � � �

� � �

, and

extend � to a cone

� � � � � �� � � ��

over the final sequence of

.

Prop. ��� � �� � � ��

is an idempotent self map of the

-coalgebra� � ��� � .

Page 17: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A final coalgebra

Write �� � � � �, where the retraction � � � �� � �

and the section

� � � � � ��

satisfy � � � � � �

. Notice that the pair

� � � � and

� � � � � �

is also a splitting of �� , thus, we have an isomorphism � making the

diagram below commute.

� �� � ���

���

���

� � � ��� �

�� � ��

� � ���

���

� �� �

��� �� � � � ��� � ���

��

Thm.

� �� � is a final�

-coalgebra.

Page 18: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A squeezing argument

We show that the injection

� � �

� � � � �� � � ��

factors through

� � � � � ��

and vice versa.

Prop. �� � � � �

� � � �

� .

Pf. Note that � � � � �� in general. We show that � � � � � ��

� � � �� foreach � � �.

� � � � � � � �

� � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� �

� � � � � � � � � �

� �

� � � � � � � � ��

� � � � � ��

� � � � �

� � �

Page 19: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A squeezing argument

� � ���

�� � ���

� �� �� � �

��� � � ��

� � � ��

��

� � � � � �� ��

� � � � ��

��� ���

��

� � � �� ��

� �

����

��

� ��� � �� �

��� �� � � � �� ��

Page 20: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A squeezing argument

� �

���

� ���

�� � ���

� �� �� � �

��� � � ��

� � � ��

��

� � � � � �� ��

� � � � ��

��� � ��

��

� � � �� ��

� �

����

��

� ��� � �� �

��� � � � � � �� ��

Page 21: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A squeezing argument

� � ���

�� � ���

� �� �� � �

��� � � ��

� � � ��

��

� � ��� � � ��� �

���

� � � � � �� ��

� � � � ��

��� ���

��

� � � �� ��

� �

����

��

� ��� � �� �

��� �� � � � �� ��

Page 22: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Main result

Thm. Let � be a regular cardinal. The class of endofunctors on

�� �

whose

final sequence converges in � � � steps is closed under

1 �-accessible functors;

2 ��� -continuous functors;

3 composition of functors;

4 arbitrary coproducts of functors;

5

-indexed-limits of functors, for any small category

.

Page 23: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A non-example

Consider the subprobability-distribution functor�� ��� � � �� �

. For a

set

,

� �

is the set of functions � � � � ��� � �with countable support

such that � � � � � � � �

. For � � � �and

� � �

define

� � � � � � � � � � . We can extend�

to an endofunctor on

��� �

by

defining, for a function

� � � �,

� � � � � � � � � � � � ��

Page 24: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

A non-example

Prop. In the final sequence

� � � �� �� �

, the map

� � � is injective.

Pf. Suppose �� � � � � � � �

and

� � �

� � � � � �

� � . Then for each

� � � � � � � �

, since � ��

(being a measure) preserves decreasing

countable intersections,

� � � � � � � � �� � �

� �� � � � � � � � � �

� � � � � � �� � � � � � � � � �

� � �� � �� � � � � � �

Similarly we get � � � � � � � � � �

� � �� � �� � � � � �

. But

� � �� � � � � �

� � � � � �� � � � � �

� � � �� � � � � �

� � � � � � � � � � � � ��

� �

for all � � �. Thus � � �.

Page 25: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Other categories

Say that

� � �

is a perfect generator if

is finitely presentable,

projective and for each object

the canonical map

��� ��� � � �� �

� � �is a bimorphism.

Examples. The one-element poset is a perfect generator in

� � �

, and the

graph with one vertex and no edges is a perfect generator in

��� � � �

.

Thm. (Adamek) If

is locally finitely presentable with a perfect

generator, and if

� � � � �is a finitary functor preserving strong monos

and bimorphisms, then the final sequence of

stabilises in � � � steps.

Page 26: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Further work

The � � � bound on the stabilisation of the final sequence of a set functor

is tight. However, we do not have an example of an � -accessible set

functor whose final sequence takes fully � � � steps to stabilise. If the

functor preserves limits of chains of monos, then one has stabilisation in

� � � steps. This case seems to cover all of the ‘reasonable’

� -accessible set functors one comes across, e.g., the countable powerset

functor.

For functors on locally finitely presentable categories, are all (any) of

Adamek’s side conditions necessary?

Page 27: On the Greatest Fixed Point of a Finitary Set Functor · 2005. 11. 30. · functor whose final sequence takes fully steps to stabilise. If the functor preserves limits of chains

Thank you for your attention!

James Worrell. On the final sequence of a finitary set functor. Theoretical

Computer Science, 338(1-3): 184–199, 2005.