lnternat. J. Math. & Math. Sci.
VoW. 9 No.l (1986) 1-16
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
RICHARD T. BUMBY
Department of Mathematics
Rutgers University
New Brunswick, New Jersey
08903
and
DANA MAY LATCH
Department of Mathematics
North Carolina State University
Raleigh, North Carolina 27650
(Received January 9, 1984)
ABSTRACT.
This paper presents some graph-theoretic questions from the
the portion of category theory which has become common knowledge.
viewioint of
In particular,
the reader is encouraged to consider whether there is only one natural category of
graphs and how theories of directed graphs and undirected graphs are related.
KEY WORDS AND PHRASES.
Category of graphs, algebraic structure.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
05C99, 18B99, 18C99.
INTRODUCT ION
The use of category theory in graph theory is no longer novel (P. Hell
[i]), and
methods of category theory have been made generally available to the ’Worklng
Mathematician".
number of MacLane
The main category theoretic topics that we need and the section
[2]
in which they may be found are:
adJoint functors (Chap. IV);
representable functors (Sect. 111.2); Kan extensions (Chap. IX); functor categories
(Sect. II.4); monads (Chap. Vl); and cartesian closed categories (Sect. IV.6).
Nevertheless, one can find examples in the literature of errors, imprecision, or
laborious constructions of special cases.
As a result of reviewing some of this work
for Mathematical Reviews, we set ourselves the task of presenting a sound description
of some of the material which seemed most susceptible to these flaws.
In particular, we wish to answer the question in the title of the paper of
Harary and Read [3] with a firm "No’.", and elaborate on defects noted in the reviews
of Farzan and Waller
[4]
and Ribenboim
[5]
(see also Ribenboim
[6]).
The concepts
of "involutorial graph" and "hom-graph" are defined by Ribenboim without citing any
earlier work. This makes it difficult to assess their pedigree. We shall present
evidence that, as originally presented, these definitions were wrong since they did
not account for the category theoretical properties which have been recognized as
essential parts of the algebraic structure and cartesian closedness.
with a study of a
"category
The difficulty
of graphs" is that graph theory was born as a branch of
R. T. BUMBY AND D. M. LATCH
2
combinatorics which leans towards defining graphs in terms of adjacency properties
of a set of vertices, which restrict the available objects and ignores the relations
between them, while category theory emphasizes the structure preserving mappings
(morphisms) and gives its best results where one is able to construct objects
"freely determined by some properties".
Our first observation is that directed graphs are easier to describe than
undirected graphs.
In Section 2, we construct categories which give satisfactory
descriptions of a category of directed graphs and which have simple categorical
The construction of a category of undirected
descriptions (as functor categories).
graphs is discussed in Section 3.
Second, vertices and edges have traditionally been considered to be different
things, but Ribenboim treated vertices as being degenerate edges.
In Section 2, we
study the relationship between the categories of directed graphs which model these
In Section 4, we observe that the "Cartesian" structure has more
intuitively satisfying properties when the vertices are considered as degenerate
two definitions.
edges.
Section 3 is devoted to recapturing an undirected graph from one of its
canonically generated directed graphs.
This process uses the theory of monads which
stems from the definition of "algebraic
structure"
2.
.
TWO POSSIBLE DEFINITIONS OF
In order to describe a directed graph
vertices and a set
E
of
in the context of category theory.
"DIRECTED GRAPH"
G, one first specifies a set
V
of
Each edge is considered as starting at a vertex,
Actually, these
(for terminal) from E to V.
called its origin and going to another vertex, called its terminal.
(for origin) and
o
assignments define functions
t
(el,e 2
This definition allows oriented graphs to have multiple
e
I
e2,
o(e I)
o(e 2)
and
t(e I)
t(e2))
(e e E with
and
e E with
o(e)
t(e)).
Often, such graphs are excluded in combinatorlal graph theory problems.
This definition is equivalent to defining a directed graph as a functor from
p (see
Figure i) to the category Ens, of sets.
the category A_
1
[011
0
Figure 1.
G[0] corresponds to
lily, the functions G(
Clearly,
V
and
1):G[l]
G[0]
The category
G[1]
and
E.
to
G(6
A
Then, without loss of genera-
0):GIll
G[0]
are the maps
and terminal, respectively.
If graphs are functors, then the appropriate definition of "graph
morphism"
should be a natural transformation between these functors. Such a
natural transformat ion
f:Gl’: G2:AP
is given by a map of vertices
f[O]:Gl[O]
G2[O]
Ens
and a map of edges
with" origin and terminal. Thus a category of
directed graphs can be modeled by the functor category [A_P,Ens].
f[l]:Gl[l]
G2[I]
which
"commute
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
3
By contrast, considering vertices to be "degenerate edges" leads one to view
the category of graphs as the functor category [BP,Ens]
(see Figure 2).
(S I
(slo.0
[o]
c0
(S O
oO(s i
ld[0
The Category B
Figure 2.
H:B_p
If
H.
edges of
Ens,
H[0]
then
represents the vertices of
H(61):H[I]
The functions
(i-- 0,1)
H[0]
and
H
H[I],
and
H(0):H[I]
HI0]
the
are again
said to be origin and terminal, respectively.
In addition, the identities
i
i
I
in
0,
insure that the function
H[I] is an
],
o0
H(o0):H[0]
_B
id[0
embedding of the set of vertices into the set of edges of
a loop based at that vertex.
H, mapping each vertex to
Thus, each vertex can be thought of as a "degenerate
edge".
[AP,Ens]
Since
and
[BP,Ens]
are both examples of functor categories, we
are able to use general properties of functor categories to describe the fundamental
category-theoretic constructions within each category, and to relate these two
candidates for a category of directed graphs.
The first principle which we use is
that limits and colimits (including the terminal object as a special case of a limit
and the initial object as a special case of a colimlt) are computed "pointwise".
For example, the null set
unique (null) map of
[Bp,Ens]
Ens
is an initial object in
into every set.
An initial object in
since there is a
[AP,Ens]
or
is a graph with no vertices, no edges, and all structural maps null.
The null graph is thus essential to a categorical approach.
Similarly, since a terminal object of
[A__P,Ens]
object of
ns
is any one pointed set, the terminal
must have one vertex and one edge which is a loop.
[BP,ns],
In
the unique edge of the terminal object T is degenerate. Thus T
the property of representing the vertex set of a graph, i.e.,
H[O] and the
has
hom-set
p_
[B_ ,Ens] (T,H)
are isomorphic as sets, for each graph
_[0]:[BP,Ens]
ns
[BP,ns] (r,_):[BP,Ens]
and
H.
ns
In fact,
are functors which are
naturally equivalent.
Since a product in a category is a limit, products in
are also computed "pointwise".
vertices
maps from
e[0]
P[I]
EXAMPLE.
vertices (e
P[0]
and
The product
P
GxG’
[AP,ns]
of A-graphs
G
G[0]xg’[0] and edges PIll G[I]G’[I]. The origin
to P[0] are constructed canonically from these maps
Suppose
from
a
G
to
and
b
and
G’
(b,b’)
[BP,Ens]
G’
has
and terminal
in G and
each have one edge connecting two distinct
e’ from a’ to b’, respectively). Then
{(a,a’),(a,b’),(b,a’),(b,b’)}; e[l]
p(0)(e,e,)
and
and
(see Figure 3).
{(e,e’)};
P(61)(e,e’)=
(a,a’);
G’
R.T. BUMBY AND D. M. LATCH
4
(a,b’)
(b,b’)
(a,a’)
(b,a’)
Figure 3.
EXAMPLE.
H[I]H’[I]
Q[I]
maps in
H
Q(60), Q(61)
with
H’.
and
H
of B-graphs
Q
The product
H
Q
Q[0]
has
H[0]H’[0]
and
constructed from the corresponding
H’
and
nondegenerate edge, then the product graph
H’
and
Q(o 0)
and
In particular, if
GxG’
The A-graph
each have two vertices and one
has four vertices and nine edges,
four of which are degenerate (see Figure 4).
(a,b’)
(b,b’)
(a,a’)
(b,a’)
Figure 4.
The B_-graph
HH’
[Ap,Ens]
Since the same construction can product very different results in
and
[Bp,fns],
it is desirable to be able to relate these t-o categories so that
-
constructions in one would be naturally "induced" in the other.
u:A
The inclusion
functor
For a B-graph
p
category A
B
of
various vertices
B_p.
v e
as a subcategory of
U(H)
induces a (forgetful)
H to the subfor
U(H), the degenerate edges
is just the restriction of
H(o0)v,
In the A_-graph
H[0],
B
[AP,ns].
U:[BP,ns]
H, the A-graph
of
A
are no longer distinguished.
The restriction of natural
transformations gives no difficulty, and it is easy to verify that
U:[BP,ns]
Because
op p
:A
u
B
[AP,ns] is actually a
U:[BP,Ens] [AP,Ens]
p,
For each A-graph
setting
is defined by composition with the inclusion
U
it induces more structure:
L
are the Kan extensions
by adding to
functor.
G[l]
L(G)(O
and
.
R.
,
G, L(G), the "B-graph freely generated by
a new degenerate edge
0)(v)
If
formation from the functor
defined to agree with
f
G
f:G
G’
is an
to the functor
when "restricted to
(e(f)[l])()
To see that
U:[Bp ns]
L:
[AP,ns]
[Ap Ens]
has both left and right adjoints which
These are described below.
[BP,ns]
one for each vertex
G",
v e
is obtained
G[0],
and
A-morphism (i.e., a natural transL(G’) is
G’), then L(f):L(G)
G"
and to satisfy
(f[0])(v).
is the left adjoint of
one shows that there is a natural one-to-one correspondence
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
(G,H):[BP,Ens](L(G),H)
for each A-graph
G
and each B-graph
5
[AP,Ens](G,U(H))
H.
This adjoint relationship may also be described in terms of the unit natural
transformation
ue:[AP,ns]
D:Id
[AP,ns]
or the counit natural transformation
<L,U>.
of the adjoint pair
[B__P,ns]
id:[BP,ns]
:LU
We now construct both natural transformations. The
B_-graph L(G) may be restricted to p to give
construction, given above, for the
UL(G).
the A-graph
A=
G
This construction yields
G:G
the collection of these inclusions
UL(G)
In fact, this
<L,U>. Similarly,
G.
is natural in
collection is the unit natural transformation of the adjunction
the counit
:LU
id
H.
for each B_-graph
is given by describing the
LU(H)
The graph
UL(G)"
as a specific subgraph of
B=-graph
H
differs from
eH:LU(H)
morphism
H
by the addition of new
degenerate edges for each vertex, the old ones having lost their distinction in
U(H).
The B-graph epimorphism
Q
new degenerate
L
added by
H
H LU(H)
is the identity on
to the original degenerate edge
Next, the adjoint relationship
<L,U>
and maps each
H(o
0)(v)
H[I].
in
may be verified by demonstrating that the
following composites are identity natural transformations:
,qU
U
U
>ULU
U
L
L
Lastly, note that neither
nor
gH
aL
-LUL
G
>L
can be an isomorphism unless
H
or
G,
respectively, is a null graph.
The right adjoint
R:[AP,Ens]
is given as follows:
R(G)[0]
<Z0,e,l
G, R(G) is the B_-graph whose vertex
e G[I] with
loops of G (i.e. edges
for each A-graph
is the collection of all
G(d I)(Z)
G(
>
0)()).
0
R(G)[I]
The set of edges
G
of edges of
vertex of
[BP,:ns]
with
0’ Z1
is the collection of all triples
loops and
G(I)( I) G(0)(1
0,ZI, respectively; i.e.
G(0)(e).
and
an edge having origin the
e
I’
(R(G))(dl)<0,e,l > 0
and terminal the vertex of
i.e.
G(60)(0) G(61)(e)
<0,e,l > are
(R(G))(d0)<O,e,l> i"
G(I)( 0)
The origin and terminal of
and
Lastly, the distinguished R(G)-loop at the R(G)-vertex
(R(G))(g0)()
q:id
<,,>.
"-RU:[BP,ns]_
H
Particularly,
i.e.
e
of
qH(e)
H
[BZP,ns]_
{H(o0)vlv
consists of the
carries a vertex
with origin
e H[0]}.
H
of the
v
H(o0)(v),
v
i.e.
and terminal
<H(g0l)e,e,H(g060)e>.
which identifies
is
<,,>;
i.e.
The unit natural transformation
which is the degenerate loop
edge
set
Clearly,
with the full subgraph of
_B---mrphlsms
B=-graph H to
H:H
RU(H).
the vertex of
RU(H)
NH(V) H(o0)(v); and H maps an
w to the edge <H(g0)v,e,H(g0)w>,
H:H
RU(H)
RU(H)
is the B__-monomorphism
generated by vertices
R. T. BUMBY AND D. M. LATCH
6
e:UR’/Id:[AP,F-ns]
The counit
of
G.
eG
The A_-morphlsm
<o,e,l >
G[I]<0,e,I
Thus
e.
is analogously defined to be a
UR(G)
The vertices of
G.
are the loops
G(O)() G(I)()
eG[O]() G(O)() and
maps such a loop to its vertex
and carries an edge
>
[Ap,Ens]
eG:UR(G)
family of A-graph morphlsms
e
e; i.e.
to the edge
"compresses" UR(G)
G
onto the full
_A-subgraph
of
G
generated by those vertices having loops.
Note that e
is an A-graph isomorphism if and only if at each vertex of G,
G
there is precisely one loop. Similarly, H is a B_-graph isomorphism exactly when
H
each loop of
H(o0)v.
is a degenerate loop
A
isomorphism between the full subcategorles
per vertex and
8
U
Therefore
R
and
induce an
of A_-graphs with exactly one loop
of B-graphs with only degenerate loops.
These ideas above illustrate the first part of the following theorem whose
second part we use later.
THEOREM 2 1 (Lambek and Rattray)
S
X
n:id -TS:X
functors with unit
)._2-y}.
ob,ly:ST(,
{ e
V:T
be a pair of adjoint
V.
:ST *id:F
{X
Fix(TS,)
induce an equivalence between
Fix(ST,)
S:X
Let
and counit
obXlx:X
(ii)
(iii)
(iv)
(v)
(vi)
(TS,r,TS)
the triple
T
_S
X_.
on
is
_
idempotent"
is a natural isomorphism;
(ST,e,ST)
the cotriple
is idempotent"
on
is a natural isomorphism"
TS:X X
ST:- F
and
and
Moreover, the following statements are
equivalent
(i)
T
Then
TS(X)}
factors through the subcategory
Fix(TS,)"
factors through the subcategory
Fix(ST,g).
with
If these conditions hold, Fix(TS,) is a reflective subcategory of
reflector the factorization (v) and Fix(ST,g) is a coreflectlve subcategory of
F
with coreflector the factorization (vi).
See Lambek and Rattray
In the example just discussed,
PROOF.
T:F-
[BP,Ens]
coreflective fail.
v
and two loops
at
RG:RG
v.
Then
RU(RG)
RG
U:[BP,Ens]
[BP,Ens] The
[AP,Ens]
is
Fix(UR,E)
to be reflective and
Consider
0,%i
Theorem i.i.
R:[AP,Ens]
is the right adjoint
Fix(RU,N)
[8],
S:X_- F__
where
G
and
condition for
[Ap,Ens]
to be
is a graph with one vertex
has two vertices
{0,I},
each with two
loops
However, RU(RG)
has four vertices, the four loops (2.1).
Thus
nRG
fails to be an
isomorphism and condition (ii) above does not hold. In fact, FIx(RU,N)
coreflective since the inclusion does not preserve colimlts.
__
is not
S:X__.._F:T
with an adjoint pair
Suppose we compose the adjunction
adjunction
composite
the
to
2.1
applies
then Theorem
M: Y
of
subcategory
_F with idempotent cotriple
In particular, if Z_ is a reflective
-":N
(MSTN,g,MSTN), then there is a reflective subcategory X_l of
Theorem 2.1 which is equivalent to a coreflective subcategory
As an illustration, let
generated by all
A=-graphs
_,
Z_l
MS:XZ:NT.
guaranteed by
of
Z_.
F 1 be the full reflective subcategory of
which have at most one loop at each vertex.
The left
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
M:[AP,Ens]_
adjoint (usually called the reflector)
N
maps each At-graph G
FI- [A__P,Ens]
between
G, the same
edges
vertices as
G
at each vertex at which
[AP,ns]
UR:[AP,ns]
UR(G
I)
NG
of
1
UR(GI)
Note that
M:[AP,Ens]
F I of the inclusion
to the quotient
A-graph G 1 with
the same
distinct vertices, but having only one loop
has loops and having no new loops.
NG
with
1
generated by the vertices of NG
to a graph
GI I’
F I, has no further effect and
yields the full subgraph
at which there is a loop.
1
is already in the subcategory
Applying
I;
hence the function
URN (GI)’ NG
MURN(G I)
this construction yields the same graph, up to isomorphism.
Iterating
I
Thus condition (vi)
of Theorem 2.1 holds, and all the other equivalent properties (1)-(v) and the
conclusions of Theorem 2.1 follow.
F
of
Particularly, the coreflective subcategory
consisting of A-graphs with exactly one loop per vertex is equivalent to
A of B-graphs with only degenerate loops. Note that
I
the reflective subcategory
and A
p
Fix(UR,g)
U:
[Bp ,Ens]
[A
Fix(RU,q)
Ens] :R.
for the original adjunctlon
Thus it is possible for
Fix(TS,q)
to be a reflective
X_ without the existence of a factorizatlon of TS:X_/ X through
(i.e. condition (v) of Theorem 2.1 fails even though part of the conclu-
subcategory of
Fix(TS,q)
sion is still true).
2
A second important subcategory
subcategory generated by the simple
A_-graph G 2
whenever
A_-graphs G with at most one edge from any
M:[Ap Ens]
2’ the left adJoint
N:F2r-- [A__UP,Ens],--
maps each
G(l)e
and
G
A__-graph
G, but with edges
having the same vertices as
G(l)e
is the full reflective
The reflector
particular vertex to another.
of the full inclusion
[A_P,Ens]
of
G(O)e G(O)e ’.
to the quotient
e
and
e’
Using an argument similar to the one for
easily seen to be idempotent.
A
follows that there exists a reflective subcategory
equivalent to a coreflective subcategory
2
[Bp,Ens]
of
2
[BP,Ens]
I’
2
is
it
which is
of
Theorem 2.1 thus establishes equivalences between subcategorles of
and
identified
Again, the cotriple on
[_A_P,Ens]
which introduce the combinatorially useful concepts of "absence of
(unnecessary) loops" or "characterization of edges by their origin and terminal
The "combinatorially interesting graphs" are extracted in a natural way
vertices".
[AP,ns]
from either of the functor categories
two possible general definitions of
or
[BP,Ens].
Thus either of the
"directed graph" leads to the same theory.
Along the way, though, we have found that the interplay of these generalizations
leads to deeper properties than one might have expected.
3.
UND IRECTED GRAPHS
In combinatorial problems, a graph is simple considered to be one-dimensional
Each graph U has a set V(U) of vertice.s, a set E(U) of edges, and
complex.
I(U)
an incidence relation
vertices.
P(G):
V(U)
Clearly, a directed
the vertices are
G[O];
E(U)
A_-graph
with each edge incident to at most two
G
can be given this structure, denoted by
G[I];
the edges are
and the incidence relation is
the collection of pairs
{(G(60)e,e),(G(61)e,e)le
A morphism
f:U
I
U
2
e
G[I]} G[O]
x
G[I].
of undirected graphs is given by a pair of functions,
V(f):V(U I)
V(U 2)
and
E(f):E(U I)
E(U2),
R.T. BUMBY AND DANA MAY LATCH
8
such that incidence is preserved; i.e.
Any
A__-graph
morphism
f:G
G
I
"induces" such a pair
2
P(G 2)
e(f) :e(G I)
commutes with origin and terminal.
of functions between undirected graphs since
Hence, this construction induces a functor
P’[AP,Ens]
from the category of A_-graphs to the category
The category
T
I(U2)-
E(f)I(U I)
(V(f)
is not easy
T
T
of undirected graphs.
depends on using, within
to describe, since it
category theory, the notation of a set with at most two elements (to describe the
incidence relation
I(U)).
This difficulty illustrates a difference between the
combinatorial and categorical viewpoints in graph theory.
The categorical approach
searches for general concepts which may be simply described and proceeds from there
to particular or special examples" while the combinatorial approach assumes that such
The "purity" of the categori-
things as multiple edges or loops will be complicated.
cal approach aside, it appears necessary to make some arbitrary choices in order to
T.
obtain a category to model
However, we are able
to avoid this difficulty by
using external relationships (which we expect to exist) between
to
T.
T forgets orientation; hence
help "internally" construct
The functor
p.[oP,ns]
P
investigate the possibility that
OU
"intuition" about the category
T
such that they agree with our
and such that there is a natural equivalence
(G,U):T(PG,U)
/[A_P,ns] (G,OU).
Every A_-graph G is determined by the sets
edges, respectively, and by the functions G(
I)
Since
[P,Ens]
it is reasonable to
[P,Ens].
[P,Ens]
in
and
has a right adjoint
O:T
Thus we search for objects
[P,Ens]
G[O]
and
G(
and
O)
(3.i)
G[I]
of vertices and
of origin and terminal.
is a functor category, the Yoneda Lemma guarantees that these sets
and functions can be naturally described in terms of representable functors; in
particular,
G[i]
[AP,ns] (A(_,[i]),G),
i
0,i
3.2
and
G[ i]
[AP,Ens] (A( ,s i) ,G)
i
0,I.
The object (actually functor)
V
-=
A=(_, O]
p
:A
which represents vertices has one vertex since
identity, and no edges since
A([I],[O])
E
ns
A([O],[O])
is empty.
A(_,[I]):Ap
consists only of the
The object
Ens
which represents edges similarly has two vertices and one edge joining them.
morphisms (natural transformations)
S- A(
,61):A(,[0])
i A(,[i])
The
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
9
and
A(_,6 ) :_([o1)-,.
__,
,(,[1)
T _=
V
to
E
edge in
m.
The subcategory
from
select the origin and terminal vertices, respectively, of the unique
B
with morphlsms
U
For each graph
(3.1) and (3.2).
T
and
A’
The set of vertices of
(OU) [0]
Since graph
V
functor, PV
is the graph in
PV
U
of
PV.
T:V
OU
-
(OU)[O]
[_._op,=,,] (v,ou)
=
T(PV,U).
E
by using both
which is naturally
P:[AP,ns]
U
T
is a forgetful
A T-morphism
to be the image of the unique vertex
is given by the set
U.
T(PE,U).
The
A==-morphism
T(PT,U):T(PE,U)
T(PV,U). The undirected graph PE
vertices; thus T(PT,U) restricts a morphism with domain
induces a map
has one edge and two
OU
is identified with the set of vertices of
OU
E
[AP, ns] (A(_, [0] ). OU)
is given by selecting a vertex of
Thus
structure of
[AP,Ens](V,OU)
is
with one vertex and no edges.
Similarly, the edge set of
and
A.
has one vertex and no edges, and
T
V
consisting of objects
T, we can determine the
in
T(PV,U):
isomorphic to
[AP,Ens]
of
is isomorphic to
to one of these vertices.
If vertex
v
U
of
is incident to edge
PE
e, then there
is a T-morphism
f :PE/U
such that the unique edge of
designated by
PE
PT:PV
PE
is mapped to
is mapped to
and such that the vertex of
e
PE
v.
We expect that this description above would determine the morphism
uniquely.
First, if
incident with it; and
second vertex of
incident with
PE.
e
is a proper edge, then
f :PE
e
U
f
e
has precisely one other vertex
e
should have this vertex as the image of the
Second, if
is a loop, then there is only one vertex
e
e; and both vertices of
PE must be mapped to
v.
v
Note that these
assignments establish a natural equivalence between
(OU)[I]
T(PE,U)
and
I(U) V(U)
x
E(U).
In addition, the terminal map
(ou)
(o)
(ou) [1]
I(U)
is given by the natural projection of
on
(ou) [o]
V(U).
To complete the description of the directed graph
OU, it is necessary
to give
the origin map
(ou) (l) (ou) [1]
(ou)
[o].
The argument given above to justify the identification of
a construction of
(OU)(I); i.e.
(OU)(61)(v,e)
Iv’;
[
if
(v’,e)
T(PE,U)
I(U), v’ #
with
I(U)
is
v
v; otherwise
This construction uses the difficult to
either one or two vertices".
naturall[ describe notion "is incident with
The existence of the adjolnt relationship <P,O>
R. T. BUMBY AND D. M. LATCH
10
demands that the construction of the origin be natural.
op
simple functor category
adjoint pair
of algebras
Ens]
of A-graphs
Hence we will exploit the
use the assumed existence of an
<P,O>, and apply the general theory of monads
F which will be a functor category model
to construct a category
[yP,Ens]
T.
for
As a first step, we give a description of the monad (or .triple) M
<T,,U>
p
which
results
from
the
assumed
existence
Ens]
of
the
[h
adjunction <P O>
in
For each h-graph
rG[O]
G[O]
Of(G)
G, TG
TG[I]
and
is described using the above discussion.
Zo
" rG[l]
G[I]’’I
(G)
where
G[I] IC( O) (e)
{e
I(G)
[hus
-
TG( O)
G[O]’G[O]
TG[O]
I) (e) },
the se__t
(3 3)
o__f loops of
G.
The universal
are used to give a description of origin
TG(
I)
to reflect the above construction of
OP:[AP,Ens]
T
TC(60)’IG[I]
G(
ns
properties of the pushout in
and terminal
Hence,
is given as the pushout
TG[O]
[AP,Ens].
is the composition of the identification
with the function
in
G( O)
(t)
while
TG(I):TG[I]
C(6 I)
in (3.4).
TG[O]
is defined similarly by interchanging
G(
O)
and
<P,O>,
The unit of the adjunction
"-r:[Ap,ns]
:id
[AP,Ens],
is given by
ld:G[O]
rG[O]
TG[O],
of diagram (3.3), for each A_-graph
TG(I)’ G
and
G.
]a:
<P,O>.
pair
V(
U)
M
T2"-T:[AP,ns]
is constructed from the counit
For each graph
Id:V(PO(U))
V(U).
U,
cu-PO(U)
The set
U
construction of the equivalence
(G,U) :T(PG,U)-leads to the function
to
I(K):
E(u):E(PO(U))
I(PO(U))
in
E(U)
PO(U)
TG( O)
G.
[AP,Ens]
:PO
is the
E(PO(U))
above discussion, just the incidence relation
The incidence relation
_
TG[1]
Clearly, from the definition of
is an A-graph morphism natural in
The muitiplicat ion
of the monad
-= Zo:G[1
rG[1
and
id:T- T of the adjoint
identifying map on vertices,
of edges of
I(U)
V(U)
PO(U)
is, from the
E(U).
A sketch of the
[AP,[ns] (G,OU)
being the projection
I(U)
E(U).
can be given as a pushout with a map
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
id
I (Pou)
(u)
(u)
,i
(3.5)
I (U)"
t
where
is a
U
E(U).
edge in
U
"twist" map which chooses the other element of I(U) over the same
In the context of the monad M, the above constructions lead to the
pushout description (3.6) for the multiplication
NG:T2G
TG:
id
TG[I]-2
T
.W’ G[I]
/
rG[l]
(3.6)
GIll.
(3.7)
t
TG[I]
t:TG[I]
where
is defined by
z
I
Z(G)
G[l
z
Note that
TG[I]
t:TG[I]
TG[O]
G[O]
id:G[O]
together with
does not define an A-graph morphism, but
The axioms for a monad (see MacLane
0
<Id’G[I]>:T2G
G
TG
on vertices
is in
[AP,Ens]._
[2], VI.I)
are now easily verified; in
p
particular, the constructions are clearly "natural in G" G e [A
Ens]
An
alsebra
<T,,>
M
for the monad
[Ap,Ens]
lying category, which is
is defined as an object
X
of the under-
in this case, together with a morphism
h:TX
X
such that diagrams (3.8) commute:
T2X
X
nX
Th
TX
h
0
Nx[O]
(3.8)
h
TX
Since
> TX
X
X
X
(TX[0]
id:X[O]
of (3.3), we construct
h:TX
X
X[O])
and
hN X
with
DX[I]:X[I
id
x
TX[I]
is the map
from the pushout (3.9):
ld
X[]
TX[I]
x
X[l]
(3.9)
[]/zl____
b
The requirement that
h:TX
X
be an A-graph morphism forces the map
b:X[l]
X[I]
X(6 l-i) :X[l]
X[O] i 0,i) Furthermore
the restriction b/Z(X) must be the "identity" inclusion (X)
X[I]. The
commuting of the first diagram of (3.8) reduces, in this case, to the statement that
to reverse orientation
(i.e.,
x(i)b
R.T. BUMBY AND D. M. LATCH
12
X[I]
b:X[l]
has
b
i.e., b
2
Id:X[l]
X[I]"
is an involution.
Thus the concept of graph with involution arises naturally from the concept of
unoriented graph; in fact, it provides an algebraic realization of this concept.
category
M
monad
T
I
in the base category.
In addition, the Kleisli construction uses the monad
T
2
which is essentially the category of free algebras.
[AP,Ens],_
same as those of
functions
The
of M-algebras always has a pair of adjolnt functors which induce the
to describe a category
but there are additional morphisms in
<fo,fl >, fo:G[O]
f
M
g’[l]
T 2.
G’[O]:fiG[1]
T
The objects of
are the
2
Any pair of
T 2.
which preserve incidence
(rather than origin and terminal) is a morphism of
In general, the category of all algebras (in our case, T I) is a terminal object
Adj(M) of adjoint pairs which induce the monad M, and the Kleisli
in the category
T 2)
construction (in our case
[2] (VI.5,
functor of MacLane
T
every algebra in
T,
The unique
is isomorphic to a free algebra in
I
Thus, T
I
categories are equivalent.
of
Adj(M).
is initial in
:T 2
Theorem 3) is both full and faithful.
T2
and
(T2)"
TI
Furthermore,
hence, the two
are, essentially, equally good models
and they are canonically chosen from all categories inducing the monad
T
The category
T
but the category
2
has other advantages.
I
M.
is perhaps closer to our original idea of unoriented graphs
Observe that
A__-graphs
[c_P,Ens])
C_-graph (object in
can be naturally extended to a
with involution
C
with
pictured
in Fig. 5.
[o]-.---.
T61 60
T0
Figure 5.
The
category
[cP,ns]
T
of algebras
U
C__-graphs
of
the set of edges fixed by
easy to show that
T
I
6I
T
2
Id
The category C
M
defined by the monad
satisfying
U(T)
e}
{elU(r)e
{elU(60)e U(61)e}
is precisely the set of all loops of
is a reflective subcategory of
I
is the full subcategory of
TI
reflector, the left adjoint to the inclusion of
[C_p,ns].
into
U.
i.e.,
It is
Construct the
[cP,Ens],_
by passing
U, having the same
vertices as U, having the same edges between distinct vertices of U, but having
(U).
U(r)e identified with e for each loop e
weaker
than isomorphism. In particular,
is
of
categories
Note that equivalence
from a given
C_-graph
the comparison
map
II:ITml
:T 2
TII
to a new C-graph
U
T
I
UI,
a quotient of
is not an isomorphism of categories, i.e., the object
However, the notion of equiva-
is not one-to-one and onto.
lence appears to be more appropriate than isomorphism for modeling a theory.
Hence, there is only one "useful" model up to equivalence, but two different
approaches to that model.
We may also form a corresponding theory for B-graphs starting from an
orientation-forgetful functor
P’:[Bp Ens]
T’
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
T
The category of algebras
M
ties to the monad
for a monad
I
[AP,Ens],
in
M’
[BP,Ens],
in
13
with similar proper-
may be described by introducing an "involution".
These algebras approximate the "involutional graphs" of Ribenboim
[5]. Of course,
if a graph admits more than one involution, each involution defines a different
algebra. (The original definition required only the existence of an involution in
the hope of defining a
subcategory of
[BP,Ens]
into the algebraic structure in Ribenboim
Ribenboim
[5,6]
[6].)
but the involution was incorporated
Curiously, the definition in
also requires that
an edge fixed by involution be degenerate.
(3.10)
The requirement (3.10) is incorporated naturally into the program developing
the monad M’ by having
H[I] play the role of the inclusion
H(oO):H[O]-
(G):’ G[I]
of (3.3) in the definition of
H[O]
the common composition
r’H(i):T’H[l]
r’H[0],
is the analog of
The category
category
H[l]
0
(Note that the mapping
[5],
T’H[I].
With this change,
:T’T’H[I]
H’
T’H(o 0)
is
(3.3); and
in the analog of
0,i, is given as in (3.4).
i
of (3.3), and
z
(3.6) and (3.7).
(cf. Ribenboim
T’H[I]
NH[I]:H[I]
Again,
T’H[I]
T’H[I]
is constructed as in
in general, is not as isomorphism
p. 159).)
T
[DP,ns],
I
M’
of all algebras for the monad
where
D
in
[BP,Ens]_
is the
is depicted in Figure 6.
61
o
[o].__.w[
60
oOO 0I
oO T 0
61 60
id
61
T60
Figure 6.
The category
D
The corresponding Kleisli construction yields a category
TI;
equivalent to
rather the free algebras in
T2
T2
which is not
are characterized by (3.10).
Thus the original definition of involutorlal graph was designed to select a subcategory of
T
I
subcategory of
equivalent to
TI
p_
[DJ
ns]
T
2.
It is also curious that (3.10) defines a
which is both reflective and coreflective
Also, consider the Kan extensions of the functors induced by the inclusions
B-D
and
A’’C.
In particular, the forgetful functor
U:[DP,ns]
and its left adjoint (i.e., left-Kan extension) induce the monad
4.
[BP,ns]
M.
CATEGORIES OF GRAPHS ARE CARTESIAN CLOSED
A benefit of viewing a category of graphs as a functor category is that any
functor category
[xP,Ens],
for small
X
is cartesian closed (see Freyd
[9],
p. 8).
The proof of this result is constructive; i.e. an algorithm is provided for com-
puting the "internal-Hom
Hom(Y,Z)
functors.
is also an object in
If
Y, Z
[xP,Ens],
are objects in
[Xp,ns],
whose evaluation at object
p
then
of
X
is defined by:
Hom(Y,Z)(p)
[xP,Ens] (X(_,p)
x
Y,Z).
(4.1)
R. T. BUMBY AND D. M. LATCH
14
Thus, for our examples, we give the representable functors
X(_,[i]):X2 p
for
A, B, C or D.
X
and
0,I
i
ns
depicts the directed graphs given
Figure
by these representable functors.
x(,[0])
C
B
A
X
id[0
6
x(_,[])
id
[0]
ld
[.]
td[t]OoO
6
Hom(Y,Z)
in
The set of vertices is given by
[AP,Ens] (A(_,[O])
Hom(Y,Z)[O]
A(_,[O])
is a graph with one vertex and no edges.
y
"coordinatewise", A(_, [0]
A(_,[I])
Y[O]
has vertices
Hom(Y,Z)[O]
The graph
to
i__
As an example of the use of formula (4.1), we construct
But
Od[o]
Representable functors
Figure 7.
[AP,Ens].
D
Since products are computed
and n_9_o edges.
Thus
ns(Y[O],Z[O]).
has one edge joining two vertices.
the same number of edges as
Y,Z).
Thus
Y, but twice as many vertices.
A(,[i])
Y
has
The structural maps
satisfy
6i(<id>,e)
Thus Hom(Y,Z)[I] is identified with
Ell], s:Y[O] Z[O], t:Y[O]
<r:Y[l]
(<6i>,ie)
the set of all triples of functions
Z[O]>
for which diagrams (4.2) commute
r
r
Y[I] Z[I]
Y
(61)
Y[0]
The origin,
Hom(Y,Z)(61),
the terminal,
Y[I]
>Z[l]
$ZZ[O](61) Y[O]y (60) Z[O]
z[60]s
-
(4.2)
t
is given by
Hom(Y,Z)(60),
<r,s,t>-s
by
<r,s,t>--t.
EXAMPLE.
Y[O]
Z[O]
If
Y
is discrete, i.e.
Y[I]
,
then
Hom(Y,Z)
has all functions
as vertices and has a unique edge joining each pair of vertices.
In general, the set of edges joining points
s,t
Hom(Y,Z)[0]
is given by
the set of functlo=s
r:Y[l]
satisfies the
property (4.1), it seems to have little relation to graph
adJoint
Z[I]
satisfying (4.2).
Although this construction
homorphisms.
Next, consider
[BP,Ens].
vertex and one edge; so that
From Figure 7, it is clear that
B(,[0])
has one
CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY
_B(_,[O])
Thus, Hom(Y ,Z)
[O]
Y.
is the set of B_-graph morphisms:
-
Hom(Y,Z)[0]
Hom(Y,Z)[I]
Z[I]- s:Y Z; t’Y
Each "edge" of
<r:Y[l]
Y
15
[BP,Ens](B(_,[0])
[BP,Ens] (Y,Z).
Y,Z)
is represented by a triple
Z>, where
s
and
t
are
B_-graph morphisms for
which (4.3) is commutative:
(4.3)
As before,
<r,s,t>-s
Hom(Y,Z)
gives the origin of
and
<r,s,t>-t
defines the terminal.
If
s
and terminal
and
t
t
are given, then the set of edges of
The definition of Horn graph in Ribenboim
[5],
given.
Z[I]
with origin
r
to
s
satisfying (4.3).
p. 165, was in this spirit.
for no apparent reason, only the restriction of
that definition.
Hom(Y,Z)
r:Y[l]
is indexed by those functions
Y(0)y[0]
However,
entered into
In Ribenboim [6], p. ii0, a totally different definition was
Again, it was an explicit construction, and again there was no claim of
naturality.
The idea was that the edges should be indexed by the functions
r:Y[l] Z[I] of our definition. The difficulty is that (4.3) determines only
s[0] and t[0], so that the vertices could only be the equivalence classes of
"graph homomorphlsms restricting
to the same functions on
vertices".
By contrast,
our construction is no more cumbersome than these two attempts, but the underly
general principle is quite simple and guarantees the Horn will have the proper
adjoint relation to cartesian product.
Our internal Hom-functor has the usual properties of Hom in
ns
in particular,
there is a composition
o:Hom(Y,Z)
The realizations of vertices in
Hom(Y,Z)[I]
as triples
<r,s,t>
Hom(W,Y)
Hom(Y,Z)[0]
Hom(W,Z).
as graph homomorphisms and of
are compatible with the composition (4.4).
The composition (4.4) induces a natural monoid structure on
invertible elements of
called
Hom(W,W)
(4.4)
Hom(W,W).
are selected, the resulting subgraph is a
If the
group
Aut(W).
The various reflective subcategorles constructed in Section 2 and Section 3 in
the discussion of categories of graphs sould be cartesian closed.
In fact, the
following proposition gives an easy computation of Hom in many cases.
R. T. BUMBY AND D. M. LATCH
16
PROPOSITION.
Suppose
F
full reflective inclusion with
pair).
in
Then
R
is a cartesian closed category and
R:F
F’
I:F’
the reflector (i.e., <R,I>
F
A
preserves products iff for all
and
B
is a
adjolnt
’, Hom(A,B)
is
F’
PROOF.
REMARK.
See Freyd
[9;
p.
13].
The reflective subcategory
2 does not satisfy this property; however
ACKNOWLEDGEMENT.
AI
A2
of
[P,Ens]_
constructed in Section
does.
We give special thanks to F. E. J. Linton for his advice during
Thanks are also due to M. L. Gardner, P. Hell and
the writing of this paper.
S. MacLane for their encouragement and useful conversations.
REFERENCES
i.
HELL, P.
2.
MACLANE, S.
1971.
3.
HARARY, F. and READ, R.C.
4.
FARZAN, M. and WALLER, D.A. Kronecker Products and Local Joins of Graphs,
Canad. J. Math. 29 (1977), 255-269, MR 55#2625.
RIBENBOIM, P. Graphs with Algebraic Structures, Report to the Algebra Group,
Queens Papers in Pure and Applied Math. 36, Queens University (1973),
155-184.
5.
An Introduction to the Category of Graphs (in Topics in Graph Theory,
New York, 1977) Ann. New York Acad. Sci. 328 (1979), 120-136.
Categories for the Working Mathematician, Springer-Verlag, New York,
Is the Null Graph a Pointless Concept?, Graph Theory
and Comblnatorics, Springer-Verlag, 1973, 37-44.
6.
RIBENBOIM, P.
105-123.
7.
MACDONALD, J. and STONE, A. Topoi Over Graphs, Cahiers Topologie Geom.
Differentiable 25 (1984), 51-62.
LAMBEK, J. and RATTRAY, B.A. Localization and Duality in Additive Categories,
Houston J. Math. i (1975), 87-100.
FREYD, P. Aspects of Topoi, Bull. Austral. Math. Soc. 7 (1972), 1-76.
8.
9.
Algebralc Structures on Graphs, Algebra Unlversalls 16 (1983),