RFC Walters with R' Rosebrugh, N' Sabadini

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Calculating limits and colimits compositionally
RFC Walters (with R. Rosebrugh, N. Sabadini)
Università dell Insubria, Como, Italy
CT2007, Faro, Portugal
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In Computer Science The state spaces of systems
are often described by finite limits or colimits
in a category E, parametrized by a graph G which
describes the geometry of the system. It is
desirable that there is also an algebraic
description, so that the limit or colimit is
described by an expression, rather than
geometrically. This goes back to the beginnings
of computer science, where - a program may
be described either by a flow chart
(gotos), or program text (while)
(Böhm-Jacopini) - a language may be
specified by an automaton or an expression
(Kleene) And of course it is present in
inumerable areas of computer science (Petri nets
versus process algebras, wysiwig versus markup,
graph versus term rewriting, etc etc) and
mathematics.
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In Category Theory Finite limits and colimits
are parametrized by graphs, ie geometrically. In
this lecture we show that they can also be
described by expressions in an algebra. As an
application we show Kleenes theorem.
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What algebra?
Assume now E has finite colimits. What is the
algebra in which finite colimits in E can be
described by expressions? Answer
cospan(E). The operations and equations of the
algebra? Answer symmetric monoidal category in
which each object has a commutative separable
algebra structure. We call such a wscc category
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(Carboni, Walters, Cartesian bicategories I, JPAA
1987)
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Cospan(E)
Each arrow f A B of E is represented in
cospan by A B
B To avoid confusion between calculations in E
and cospan(E) I will always write cospan
calculations in a blue rectangle.
f
1
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Before stating the theorem lets look at an
example How is a coequalizer in E given as an
expression in this algebra? Consider f, g A
B the coequalizer of f and g is the centre
of the cospan A
AA BB B

fg
f
g
8
fg
A
AA
B
BB
A AA BB
B A BB
B
fg
1
1
1
pushout
coequal(f,g)
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The Theorem
Let Graph be the category of finite graphs, let
E be the underlying graph of E. Consider
Graph/E, the category with objects diagrams in
E, and morphisms compatible graph
morphisms. Then colim is a functor colim
Graph/E E. Consider
Cspn(Graph/E) the full subcategory of
cospan(Graph/E) with objects being discrete
diagrams in E. Theorem Colim extends to a
functor colim Cspn(Graph/E)
cospan(E) which preserves the
wscc structure.
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Theorem. Colim extends to a functor
colim Cspn(Graph/E)
cospan(E) which preserves the wscc structure.

Remarks. Cspn(Graph/E) is the result of freely
adding wscc structure to the graph E.
(Rosebrugh,Sabadini, Walters - CT04) This means
that diagrams in E may be written as expressions
in the wscc structure of Cspn(Graph/E) with
constants being the arrows of E. Then colim
preserves wscc expressions, so the colimit of any
diagram may be written as an expression in
cospan(E).
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Example of coequalizers
f
The diagram A B is given the by
evaluation in Cspn(Graph/E) of expression

g
fg
AA
B
BB
A
Hence colim(f,g A B) coequal(f,g) is
given by evaluating the same expression in
cospan(E).
Note In Cspn(Graph/E)
A AA is the cospan of diagrams
A A
A
A
And fg is the disjoint union f two arrows
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h
Example
B
g
k
f
The diagram
A
C
is given by the following expression in
Cspn(Graph/E) 0 BB
BAABB
BAABBB
BABABC
BAABBC BBC C 0

BAAB
B ?
?
BAfghk
Be C
eC
twist
Note ? ! e !
f
g
Hence the colimit is given by the same expression
in evaluated in cospan(E)
h
k
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Proof of Theorem

• The main point to check in showing that colim is
  a monoidal functor is (a special case of) the
  following
• Consider a diagram D of diagrams in E
  parametrized by a graph G
• i.e. a graph morphism D G Graph/E.
• We can do two things
• Calculate first the colimit of D in Graph/E to
  obtain a diagram in E of which we may then take
  the colimit in E,
• that is calculate colimE(colimGraph/E(D))
  
• 2) Calculate the colimit of
• G Graph/E
  E,
• that is, calculate colimE(colimE ? D).
• Lemma colimE(colimGraph/E(D)) colimE(colimE ?
  D).

colim
D
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Lemma colimE(colimGraph/E(D)) colimE(colimE ?
D).
Sketch of proof Suffices to show a bijection
cocones colimGraph/E(D)
X cocones colimE ? D
X But it is not hard to show that both of
these are equivalent to a compatible family
of cocones D(g)
X (g in G).
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Very special case To show a bijection
cocones colimGraph/E(D) X
cocones colimE ? D
X Diagram of diagrams is Then
colimGraph/E(D) is And a cocone to X consists
of 3 arrows A X, B X, C X Further
colimE ? D is And a cocone to X also consists of
3 arrows A X, B X, C X We have just
proved that ABC A(BC).
B C
A
A B C
A BC
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The meaning of the lemma is that one can form the
colimit of a composed diagram knowing the
colimits of the parts. Just as in the special
case this implies a variety of associativities,
which arise from the fact that a diagram may be
decomposed in various ways. There is a similar
result for limits, which however also involves
colimits of diagrams.
Lemma limE(colimGraph/E(D)) limE(limE ? D).
Limits may be calculated compositionally in
span(E).
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Example of theorem
A general diagram may be written in
Cspn(Graph/E) by an expression of the following
form
S(codomains)
d1
S(arrows)
S(domains)
S(all vertices)
d0
Evaluated in cospan(E) this translates into the
coequalizer of two arrows of E given by the top
and bottom rows. The two arrows are
S(domains) S(codomains)
S(all objects) and
S(domains)
S(all objects) the classical
formula for colimits in terms of sums and
coequalizers.
S(arrows)
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Limits and colimits of monoidal diagrams
Systems in computer science are ususally not
constructed from parts with one input and one
output, like arrows in a graph. Components have
multiple inputs and outputs, i.e. they are arrows
in a monoidal graph. Definition A monoidal graph
consists of a set V of vertices, and a set A of
arcs and two functions d0,d1A V (the free
monoid on V). There is an obvious notion then of
monoidal diagram in a monoidal category. Definiti
on Let E be a category with finite colimits,
regarded as a monoidal category with sum as
tensor. The colimit of a monoidal diagram is an
object C with a family (qi Ai C) (Ai objects
of the diagram) such that for any arrow in the
diagram fA1A2Am B1B2
Bn (q1 q2 q3 qn) f (q1 q2 q3 qm),
and is universal such.
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Monoidal colimits are also calculable
compositionally, in the algebra cospan(E). We
look at one example only. Example Consider the
monoidal diagram in Sets with three objects A,B,C
and one arrow f AC BC. It is
convienient to draw this as a flow chart
A
B
f
C
This diagram is given by an expression in
Cspn(MonGraph/E) as follows A
ACC BCC
B
B e
fC
A ?
where the centre of the cospan f is the monoidal
diagram with four objects A,B,C,C and one arrow f
AC BC.
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A
Example
B
f
So the colimit can be calculated in cospan(Sets)
as A ACC
BCC B
A ?
B e
fC
Or in Sets as
A ACC BCC
B AC
BCC BC
fC
B
BCC
A
inc
inc
pushout
The colimit consists of orbits of ABC under f
The pullback of the resulting cospan is the
partial function obtained by iterating f
colim
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Kleene Theorem the languages recognized by
finite state automata are the closure of
singletons under union concatenation and
The category E we consider is P(S)-Cat,
categories enriched in S languages. There is a
wscc functor Cspn(Graph/ S)
Cspn(Graph/P(S)-Cat)
cospan(P(S)-Cat) which takes a labelled graph
(with input and output states) to the (P(S)-
category whose homs are the languages traced out
from the domain to the codomain. This is already
a Kleene type theorem since the left-and side is
generated as a wscc category by single labelled
edges, and hence the image on the right-hand side
is also generated as a wscc category by singleton
languages. However it is not the classical
Kleene theorem, since the right hand side does
not consist of single languages and the wscc
operations are not the Kleene operations. Further
the functor does not lose internal states.
CT04
colim
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To obtain a theorem closer to the classical
Kleene theorem we compose with a further wscc
functor cospan(P(S)-Cat)
corel(P(S)-Cat) which uses the
bijective-on-objects fully-faithful factorization
to obtain from a cospan of P(S) categories a
corelation of P(S) categories. The final
composite Cspn(Graph/ S)
corel(P(S)-Cat) takes a
labelled graph with initial and final states to
the category with objects only the initial and
final states, and homs the languages traced
out. To finish a proof of the classical Kleene
theorem we need to show that the wscc operations
in corel, at the level of languages (homs), may
be expressed in terms of union, concatenation and
.
F
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To show that the wscc operations in corel, at the
level of languages (homs), may be expressed in
terms of union, concatenation and . Clearly the
tensor does not change the languages. The problem
is the composition. But in a wscc category the
general composition reduces to considering the
very special case of the colimit identifying two
objects in a category. This amounts to the
particular problem of passing from the
category
to
c
c
b
a
b
a
d
d
But the category generated by the right-hand side
has hom (aUbUcUd). This proof is very close to
one of the usual proofs of Kleene (if you strip
the superstructure). The superstructure has the
advantage of suggesting needed generalizations,
e.g. to parallelism.
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Comments The theorem we have described concerns
calculating colimits as objects, not as functors.
So we have not shown the compositionality of
morphisms. We believe this is connected with the
algebra of span and cospan as monoidal
bicategories rather than categories. We have made
an initial progress in understanding this
question, by considering a very special case,
where we identify the role of 2-separable
object. A universal property of the monoidal
2-category of cospans of finite linear orders and
surjections, M. Menni, N. Sabadini, R. F. C.
Walters http//arxiv.org/abs/0706.1393v1