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 WorrellOxford University Computing Laboratory, UK
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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.
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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.
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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
�
.
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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.
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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.
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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.
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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
�
.
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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.
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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.
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Growing trees
Below we draw a row of trees taken respectively from
� � �� � �� � � �
and� � �
, with each tree projecting down to the tree to the left.
� � �
��
��
��
�
�
��
��
��
�
��
��
��
�
� � � � � �
� � ��
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In the limit
We can imagine the trees as projections of the following compactlybranching infinite-depth tree in
� � �
.
�
��
��
��
��
��
��
��
��
�
���������������������
� � � � � � �
� � �
� �� � �
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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
�� � �� � � � ��� � � � � � � � � � ��
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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.
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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 � � � � � � � � � �.
��
��
� ��� � �
� ���
� ���� � � �
� � ���
� � ���� � � �
� � ���
� � ���� � � �
� ���� � ���� � ���� � � ���
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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� � ��� � .
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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.
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A squeezing argument
We show that the injection
� � �
�
� � � � �� � � ��
factors through
� � � � � ��
and vice versa.
Prop. �� � � � �
�
� � � �
� .
Pf. Note that � � � � �� in general. We show that � � � � � ��
� � � �� foreach � � �.
� � � � � � � �
�
� � � � � � � � � � � �
�
� � � � � � � � � � � � � � � �
� �
� � � � � � � � � �
� �
� � � � � � � � ��
� � � � � ��
� � � � �
� � �
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A squeezing argument
� � ���
�� � ���
� �� �� � �
��� � � ��
� � � ��
��
� � � � � �� ��
� � � � ��
��� ���
��
� � � �� ��
� �
����
��
� ��� � �� �
��� �� � � � �� ��
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A squeezing argument
�
� �
���
� ���
�� � ���
� �� �� � �
��� � � ��
� � � ��
��
� � � � � �� ��
� � � � ��
��� � ��
��
� � � �� ��
� �
����
��
� ��� � �� �
��� � � � � � �� ��
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A squeezing argument
� � ���
�� � ���
� �� �� � �
��� � � ��
� � � ��
��
� � ��� � � ��� �
���
� � � � � �� ��
� � � � ��
��� ���
��
� � � �� ��
� �
����
��
� ��� � �� �
��� �� � � � �� ��
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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
�
.
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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
� � � �,
� � � � � � � � � � � � ��
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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 � � �.
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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.
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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?
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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.