Adjoining adjoints

http://www.elsevier.com/locate/aim

Advances in Mathematics 178 (2003) 99–140

Adjoining adjoints$

Robert Dawson,a,* Robert Pare,b and Dorette Pronkb

aDepartment of Mathematics and Computing Science, Saint Mary’s University,

Halifax, NS, Canada B3H 3C3bDepartment of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada B3H 3J5

Received 11 March 2002; accepted 30 April 2002

Communicated by Ross Street

Abstract

We construct the 2-category obtained from a category by freely adjoining a right adjoint for

each morphism and isolate its universal property. Some others basic properties are also

studied. Some examples in which the category is freely generated by a graph are discussed in

detail. For these categories, the 2-cells are given a geometric interpretation and shown to be

similar to certain diagrams which have appeared in the literature on Cn-algebras.

r 2003 Elsevier Science (USA). All rights reserved.

MSC: 18A40; 18D05

Keywords: Adjoint; 2-Category; Localization; Kauffman diagram

0. Introduction

The usefulness of Span in bicategory theory is well documented. For a category Cwith pullbacks, the bicategory SpanðCÞ is characterized by the property that it is theresult of freely adding right adjoints to the arrows in C subject to the Beck condition(cf. [5]). In this paper we describe a 2-category P2C; which is the result of freelyadding right adjoints to all arrows in C; for any category C: This P2-constructioncan be viewed as an extension of Span in the sense that one does not need thepresence of pullbacks in C to apply it. For categories with pullbacks the two

ARTICLE IN PRESS

$Authors are supported by NSERC Grants.

*Corresponding author.

E-mail addresses: [email protected] (R. Dawson), [email protected] (R. Par!e), pronk@math-

stat.dal.ca (D. Pronk).

0001-8708/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0001-8708(02)00068-3

constructions will yield different results, but there are interesting relations, which willbe spelled out in [5].One example of a 2-category of the form P2C is found in the paper [12], where the

category Adj ¼ P2 2 is described in full detail (although the primary goal of theauthors was to study monads).The 2-category P2C is a localization of the category C; and in many ways it

resembles the hammock localization as defined by Dwyer and Kan [6], in the sensethat the 2-cells in P2C correspond to connected components in the 2-cell category ofthe hammock localization. The details of the relationship between P2C and some ofthe other localizations of a category will be the subject of another paper.The 2-cells of P2C are defined as equivalence classes of a type of diagram which

we call fences. It is not unusual for there to be some kind of equivalence relationinvolved in the definition of a localization, especially for the top dimensional cells;see for example the category of fractions as defined in [7] or the bicategory offractions as defined in [11]. However, the equivalence relations in those localizationsare all decidable if there is enough information about the original category. This isnot the case for P2C as we have shown in [3,4].We call the result of our construction P2C; the fundamental 2-category of C;

because the morphisms are paths—zig-zags of arrows—in the original category, andthe 2-cells resemble homotopy classes of homotopies between these paths. The fulldetails about the correspondence between this category and the homotopy groupP2ðBCÞ of the classifying space will also be the topic of another paper.The extension of the P2-construction to 2-categories and more generally,

bicategories will be spelled out in a forthcoming paper. If C is a 2-category, theresulting P2C will again be a 2-category. When C is only a bicategory, the result willof course also be a bicategory.We end this paper by discussing the result of applying this P2-construction to a

category which has been freely generated by a graph. In this case the equivalencerelation on the 2-cells is decidable. Moreover, they can be represented graphically bydiagrams, related to the Feynman diagrams of particle physics and to the Kauffmandiagrams used in operator theory. In future research the authors plan to investigatethe relationship between categories, Feynman diagrams, Kauffman diagrams, and

Cn-algebras more closely.

1. The free adjoint construction

1.1. Motivation

Recall that for a pair of arrows ðA-f

B;B-g

AÞ in a 2-category C; the arrow f is

left adjoint to g (and g is called right adjoint to f ) when there are 2-cells Z : IA ) gf

and : fg ) IB satisfying the triangle identities, f � f Z ¼ if and g � Zg ¼ ig: It

follows from these identities that (right) adjoints are unique up to canonical 2-isomorphism. So if one wants to add right adjoints to the arrows of a (2-)category in

ARTICLE IN PRESSR. Dawson et al. / Advances in Mathematics 178 (2003) 99–140100

a free way, it is sufficient to add a morphism in the opposite direction for each arrow

A-f

B in C:We will denote this morphism by a backward arrow with the same label:

B’f

A: ð1Þ

Thus, one obtains zig-zags of arrows of C; as in any localization of a category. Thedifference from other localizations will be in the way we define our 2-cells. Before weexplain how to construct the 2-cells, we want to consider the shape of the zig-zagsmore carefully; there are two issues.First, we want the embedding of the category C into the 2-category P2C to be a

2-functor rather than only a bifunctor. In particular, we want it to preserve identityarrows. This could be done in two ways: one can take isomorphism classes of zig-

zags, and require that the arrow A-IA

A is equivalent to the empty zig-zag, or one can

restrict the shape of the zig-zags. Moreover, for composable arrows A-f1

B-f2

C and

chosen adjoints f n1 ; f n

2 ; and ðf2f1Þn; the arrows f n1 f n2 and ðf2f1Þn are isomorphic, but

not necessarily equal. However, the notation of (1) suggests composability.We can resolve both of these issues by choosing the morphisms of the 2-category

P2C to be zig-zags of the form

X -f0

B0’g1

A1?An -fn

Y ;

i.e., beginning and ending with a forward arrow. Such zig-zags are composed by

concatenation together with composition in C: For example, X -f0

A’g

B-f1

Y

composed with Y -p0

C ’q

D-p1

Z is X -f0

A’g

B -p0f1

C ’q

D-p1

Z: The right adjoint

f n of an arrow A-f

B then becomes

B-IB

B’f

A-IA

A:

The composition of the adjoints C -IC

C ’f2

B-IB

B and B-IB

B’f1

A-IA

A is

C -IC

C ’f2

B-IB

B’f1

A-IA

A; ð2Þ

which is not the same as

C -IC

C ’f2f1

A-IA

A: ð3Þ

Note that one could make different choices, e.g., take equivalence classes of zig-zags, or let the embedding be only a bifunctor, and the result would be an equivalent

ARTICLE IN PRESSR. Dawson et al. / Advances in Mathematics 178 (2003) 99–140 101

2-category or bicategory P2C; although the amount of bookkeeping needed to provethis may vary.We will now discuss what the 2-cells in the category P2C ought to be. Recall that

in the calculus of fractions as defined by Gabriel and Zisman [7], two morphisms

A’w

C -f

B and A’v

D-g

B are equivalent if they are part of a diagram

with uAW (the class of arrows to be inverted). Inverses are a special type of adjoints,with unit and counit cells that are identities; and the 2-cells of our construction willbe similar to these diagrams. For example, the identity 2-cell for a zig-zag

X -f0

B0’g1

A1yAn -fn

Y

is of the form

as in Dwyer and Kan’s hammock localization (cf. [6]). However, we prefer to denotethis 2-cell by a diagram where we repeat the domain X and the codomain Y on thetop and bottom row,

because the tops and bottoms of our (non-identity) hammocks do not need to havethe same length, and this notation gives us more space. (In the hammock localizationone adds identities as needed to make the top and bottom the same length, because(for instance) (2) and (3) are equivalent arrows. In our construction we have‘forgotten’ the identity structure of the original category C; so (2) and (3) are notequivalent.) We will use the jj-notation only for the identities at the beginning andend of the fence, and use (e.g.) IA everywhere else. We will call such a diagram afence.To further motivate our definition for 2-cells, we consider the unit and counit for

an adjunction. We need 2-cells from A-IA

A to A-f

A’f

B-IB

B and from


B-IB

B’f

A-f

B to B-IB

B: These will have the following shapes:

ð4Þ

When one performs horizontal composition, one composes the domain andcodomain of the fence as before; thereafter some of the vertical arrows in the fencemay also need to be composed with arrows in the domains and codomains, so thatthey do not begin or end in the ‘middle’ of an arrow in the domain or codomain zig-zag. For example, horizontal composition of the fence for in (4) with an arrowðgÞ : B-C gives:

We further illustrate our definition of horizontal and vertical compositionfor fences by showing that these units and counits satisfy one of the triangleidentities, f � f Z ¼ if (the other triangle identity is proved below in Lemma 1).

Horizontal composition of the fences for Z and with f produces the following newfences:

For the vertical composition, we write one fence below the other and compose thevertical and diagonal arrows that are composable:

ð5Þ

We will call the structure consisting of objects, zig-zags, and fences, withhorizontal and vertical composition as given, GC: As the observant reader may havenoticed, this does not always satisfy the middle-four interchange law (and hence isnot a 2-category); see, for instance, Example 1 below. This should not come as a


surprise, given that the hammock localization has non-trivial 3-cells (cf. [6]). Theobjects and arrows of GC are the correct ones; we shall see that the 2-cells of P2C areobtained as equivalence classes of those of GC:

1.2. Definitions

Let C be any category.

1. The objects of GC (and of P2C) are the same as the objects of C:2. The arrows of GC (and of P2C) are zig-zags in C:

A ¼ A0-f0

B0’g1

A1-f2

B1’?-Bn�1’gn

An -fn

Bn ¼ B ð6Þ

We denote such an arrow by

ðg1;y; gn; f0;y; fnÞ;

except when n ¼ 0; in which case we will write ðf0Þ: Composition of arrows isdefined by concatenating the zig-zags and composing the two composablemorphisms:

ðg1;y; gn; f0;y; fnÞ3ðh1;y; hm; k0;y; knÞ

¼ ðh1;y; hm; g1;y; gn; k0;y; kn�1; f0kn; f1;y; fnÞ

3. A 2-cell ðg1;y; gn; f0;y; fnÞ ) ðg01;y; g0

m; f 00;y; f 0

mÞ of GC is a particular kindof diagram, called a fence. An example of a fence is

Specifically, a fence is determined by a pair of order-preserving index functions

f : f0;y;mg-f0;y; ng and c:f0;y; ng-f0;y;mg

together with families of arrows (which we will call vertical, even when they areslanted in the diagram)

kj : Bj-B0cðjÞ and hi :AfðiÞ-A0

i:

These data are subject to the following conditions:(a) jBc (i.e., cfðjÞXj and fcðiÞpi) (note that this means that the vertical

arrows do not cross, so that the resulting diagram is planar);


(b) all resulting squares commute, i.e.

f 0j hj ¼

g0jþ1hjþ1 if fðjÞ ¼ fðj þ 1Þ;

kfðjÞffðjÞ if fðjÞafðj þ 1Þ

(

for j ¼ 1;y;m;

kigiþ1 ¼kiþ1fiþ1 if cðiÞ ¼ cði þ 1Þ;g0cðiÞþ1hcðiÞþ1 if cðiÞacði þ 1Þ

(

for i ¼ 1;y; n;(c) h0 ¼ idA and kn ¼ idB:

We denote such a fence by

ðf;c; k0;y; kn; h0;y; hmÞ or ðf;c; ðkiÞ; ðhjÞÞ: ð7Þ

Note that in the example above, cð0Þ ¼ 1; cð1Þ ¼ 1 and cð2Þ ¼ 3; andfð0Þ ¼ 0; fð1Þ ¼ 0; fð2Þ ¼ 2; and fð3Þ ¼ 2:

We will now describe horizontal and vertical composition of fences.

1.3. Vertical composition of fences

When we want to construct the vertical composition of two fences

a ¼ ðf;c; ðkiÞ; ðhjÞÞ : ðg1;y; gm; f0;y; fmÞ ) ðg01;y; g0

n; f00;y; f 0

nÞ

and

b ¼ ðf0;c0; ðk0iÞ; ðh0

jÞÞ : ðg01;y; g0

n; f 00;y; f 0

nÞ ) ðg001;y; g00

p ; f 000 ;y; f 00

p Þ;

the resulting fence needs to contain a vertical arrow starting at each source Bi: Thefence a provides a vertical arrow ki : Bi-B0

cðiÞ: Then the fence b contains an arrowk0cðiÞ : BcðiÞ-B00

c0cðiÞ; so the required vertical arrow in the vertical composition of the

fences can be obtained as

k0cðiÞki : Bi-B00

c0cðiÞ:

Analogously, we can construct the vertical arrows into the A00j by backtracking in

two steps

h0jhf0ðjÞ :Aff0ðjÞ-Aj:


For example, composing

with

gives

In general, the index functions of the new 2-cell are the compositions of the indexfunctions of the original 2-cells, and the vertical arrows in the fences are thecompositions of the corresponding vertical arrows:

ðf0;c0; ðk0iÞ; ðh0

jÞÞ � ðf;c; ðkiÞ; ðhjÞÞ ¼ ðff0;c0c; ðkcðiÞkiÞ; ðhjh0f0ðjÞÞÞ:

1.4. Horizontal composition of fences

This composition is defined as an extension of the composition of zig-zags, i.e., byconcatenation, together with some compositions in the category C: This is a tensor-like construction over C: For example, when the first fence ends with the object Bn

and there is a vertical arrow ki :Bi-Bn; where ian; this arrow needs to be composedwith the first forward arrow of the codomain zig-zag of the second fence, as in thehorizontal composition of


with

gives

Unfortunately, the bookkeeping for the formal description of this process is a bitcomplicated. (This is the place where we have to pay for our choices of morphisms inP2C:) In general, if

½ %f; %c; ð %kiÞ; ð %hjÞ : ðg1;y; gn1 ; f0;y; fn1Þ ) ðg01;y; g0

m1; f 00;y; f 0

m1Þ

and

½ *f; *c; ðkiÞ; ðhjÞ : ðq1;y; qn2 ; p0;y; pn2Þ ) ðq01;y; q0

m2; p01;y; p0

m2Þ

are 2-cells such that codðfn1Þ ¼ domðp0Þ (i.e., that are composable), then the

horizontal composition is defined as ½ *f; *c; ðkiÞ; ðhiÞ ½ %f; %c; ð %kiÞ; ð %hiÞ ¼ ½f;c; ðkiÞ; ðhjÞ ;where the index functions c:f0;y; n1 þ n2g$f0;y;m1 þ m2g : f are defined by

fðjÞ ¼%fðjÞ for jpm1;

*fðj � m1Þ þ n1 for j4m1;

(

cðiÞ ¼%cðiÞ for ion1;

*cði � n1Þ þ m1 for jXn1;

(

and the vertical arrows are defined by

ki ¼

%ki if ion1 and %cðiÞom1;

v01 %ki if ion1 and %cðiÞ ¼ m1;

ki�n1 if iXn1;

8><>:

hj ¼

%hj if jpm1;

hj�m1fn1 if j4m1 and *fðj � m1Þ ¼ 0;

hj�m1if j4m1 and *fðj � m1Þ40:

8><>:


Although GC is not a 2-category, it should be noted that the triangle identities

f n � f nZ ¼ if n and f � Zf ¼ if are unambiguous, even in the absence of the middle-

four interchange law. Therefore, we can define adjunctions in the usual way (exceptthat adjoints may not be unique).

Lemma 1. In GC; every arrow of the form ðf Þ :A-B is left adjoint to f n ¼ðIB; IA; f Þ : B-A:

Proof. The adjunction 2-cells and Z are given in (4). It was shown in (5) thatf � Zf ¼ if : Moreover,

1.5. The equivalence relation

The structure GC does not satisfy the middle-four interchange law for 2-cells:

Example 1. Consider the following fences:


When one first takes all the horizontal compositions, and then the verticalcompositions, the result is

ð8Þ

When one first takes the vertical compositions of the four 2-cells in the center, andthen the horizontal compositions of the resulting 2-cells and finally the verticalcompositions with the top and bottom 2-cells, the result is

ð9Þ

This example shows that middle-four interchange law is not satisfied, but itmotivates the equivalence relation defined below, in which we make two fencesequivalent if they are related as (8) and (9) are. In the next two sections, we will showthat this is functorial, and that it is the smallest equivalence relation that turns GCinto a 2-category.

Definition 1. Two fences

y ¼ ðc;f; k0;y; kn; h0;y; hmÞ

and

o ¼ ð *c; *f; k0;y; kn; h0;y; hmÞ

are directly equivalent if there exist indices i0Af0;y;mg and j0Af0;y; ng such thatcði0Þ ¼ j0; *cði0Þ ¼ j0 � 1 and *cðiÞ ¼ cðiÞ for all other values of i; and *fðj0Þ ¼ i0 þ 1


and *fðjÞ ¼ fðjÞ for all other values of j: Furthermore, ki ¼ ki whenever iai0; hj ¼ hj

whenever jaj0: This means that the fences y and o are the same except that the cell ycontains

where the cell o contains

and there exists an arrow l :Bi0-A0j0which factors both diagrams i.e., lfi0 ¼ hj0 ;

f 0j0

l ¼ ki0 ; g0j0

l ¼ ki0 ; and lgi0þ1 ¼ hj0 : Note that the ‘direct equivalence’ relation is not

transitive (or reflexive).

Definition 2. The relation ‘B’ is the equivalence relation generated by the ‘directequivalence’ relation. The 2-cells of P2C are equivalence classes of fences under thisrelation.

It will be shown below that horizontal and vertical composition are well definedwith respect to this equivalence relation. However, as was shown in [3,4], for somecategories C; the problem of determining whether two fences are equivalent isundecidable. In some cases—for instance, when C is locally finite, or obeys certaincancellation conditions—the problem is decidable; however, there are many familiarcategories, such as Set, Top, and Grp, for which it is undecidable.

2. Composition of 2-cells

In this section we will show that the horizontal and vertical composition of fencesis well defined on equivalence classes of fences as defined in Definitions 1 and 2. Wewill also show that the class of 2-cells for P2C is minimal in the sense that it isgenerated by the units, counits, and identities.

Lemma 2. Horizontal composition of 2-cells is well defined on equivalence classes.


Proof. If two 2-cells a and b are directly equivalent via a factorization l; this samearrow l will also factorize the appropriate squares after horizontal composition witha 2-cell g; making the resulting composites directly equivalent. &

Lemma 3. Vertical composition of 2-cells is well defined on equivalence classes.

Proof. Let

o ¼ ðco;fo; ðkoi Þ; ðho

j ÞÞ

and

y ¼ ðcy;fy; ðkyi Þ; ðhy

j ÞÞ

be two representatives for 2-cells with a common domain and codomain.To prove that the vertical composition is well defined on 2-cells, it is sufficient to

prove that, if o is directly equivalent to y as in Definition 1, then, for any two fencesg1 and g2; og1Byg1 and g2oBg2b whenever these vertical composites are defined.Suppose that o contains

where y contains

and there is a factorization l : B0i-A00

j such that lf 0i ¼ ho

j ; koi ¼ f 00

j l; hyj ¼ lg0iþ1; and

g00j l ¼ ky

i as in Section 1.5.

Let g1 ¼ ðc;f; ðkgÞ; ðhgÞÞ and let n be such that fði þ 1Þ ¼ fðiÞ þ n:If n ¼ 0; the relevant parts of the composing fences are:


and

So the resulting indexing functions for the composites g1o and g1b will be the sameand the only arrows in the families of indexed arrows for which we need to check

equality are hg1oj and h

g1yj : We find that h

g1yj ¼ hy

j hg1iþ1 ¼ lg0iþ1h

g1iþ1 ¼ lf 0

i hg1i ¼ ho

j hg1i ¼

hg1oj :

For nX1; the part of g1 that composes with these parts is of the following form:

In order to keep the diagrams manageable, we will work out the case where n ¼ 2;the general case is a straightforward generalization. For n ¼ 2; the part of g1 we areworking with is of the form

The relevant part of og1 is

ð10Þ

where h0j ¼ hoj h

gi ; and k0fðiÞ ¼ ko

i kgfðiÞ:


The relevant part of yg1 is

ð11Þ

where k2fðiÞ ¼ kyi k

gfðiÞ; kn

fðiÞþ1 ¼ kyi k

gfðiÞþ1; and hn

j ¼ hyj h

giþ1:

In general, it will take n direct equivalences to show that og1 and yg1 areequivalent. For n ¼ 2; first notice that the arrow

lkgfðiÞ : BfðiÞ-A00

j

factors the square in og1: This makes og1 directly equivalent to the fence where part(10) is replaced with

ð12Þ

and

k1fðiÞ ¼ kyi k

gfðiÞ;

k1fðiÞþ1 ¼ koi k

gfðiÞþ1;

h1j ¼ lkgfðiÞgfðiÞþ1:

Now the arrow lkfðiÞþ1 : BfðiÞþ1-A00j factors both the second square in diagram

(12) and the square in yg1; so yg1 is directly equivalent to this last fence, andtherefore equivalent to og1:The proof that g2oBg2y is similar to this one, but the focus is on the B’s, rather

than the A’s. &

Proposition 1. The 2-cells in P2C are generated, through horizontal and vertical

composition, by the identity 2-cells if and if n ; the units Zf ; and the counits f ; for fAC1(as in (4)).

Proof. We will prove this by induction on the length of the fence. The shortest fencesare the ones with one right arrow on the top and the bottom. These are identity


2-cells and hence satisfy the requirements. All other fences are of one of the followingforms:

ð13Þ

ð14Þ

ð15Þ

A diagram of the form (13) can be written as the horizontal composition of thefollowing cells:

composed vertically with

This last fence is shorter than the one in (13), so it can be decomposed as required bythe induction hypothesis.A fence of the form (14) can be written as the horizontal composition of the

following cells:

ð16Þ


composed vertically with the horizontal composition of the following cells:

Note that the left-hand fence here is again shorter than the original one, so we mayapply the induction hypothesis.A fence of the form (15) can be written as the horizontal composition of the

fences in (16), composed vertically with the horizontal composition of the followingfences:

Once more, the induction hypothesis can be applied to the left-hand fence. &

Note that the proof of this proposition effectively constitutes an algorithm forfactoring 2-cells into units, counits, and identities. Equivalent fences will give rise tothe same sets of units and counits, but the identity cells may differ.

3. The middle-four interchange law

In this section we finish our proof that P2C is a 2-category by showing that itsatisfies the interchange law for horizontal and vertical compositions. We will alsoshow that the middle-four interchange law forces the equivalence relation B onfences.

Theorem 1. The horizontal and vertical compositions for equivalence classes of fences

in P2C; as defined on representatives in Section 1 and 2, satisfy the middle four

interchange law.

Proof. Suppose that

a ¼ ½f;c; ðkiÞ; ðhjÞ :ðg1;y; gl ; f0;y; flÞ ) ðq1;y; qm; p0;y; pmÞ;

b ¼ ½fw;cw; ðkwi Þ; ðh

0wj Þ :ðg

w1;y; gw

L; f w0 ;y; f w

LÞ ) ðq01;y; q0

M ; p00;y; p0

MÞ;

g ¼ ½fz;cz; ðkzi Þ; ðhzj Þ Þ :ðq1;y; qm; p0;y; pmÞ ) ðsz1 ;y; szn ; r

z0 ;y; rzn Þ;


d ¼ ½f0;c0; ðk0iÞ; ðh0

jÞ Þ :ðq01;y; q0

M ; p00;y; p0

MÞ ) ðs01;y; s0N ; r00;y; r0NÞ;

and codðpmÞ ¼ domðp00Þ; so that ðd3gÞ � ðb3aÞ and ðd � bÞ3ðg � aÞ are well defined

in P2C: The index function c must map l to m; and f0 must map 0 to 0(cf. Section 1.2). The proof of the following lemma, which we will need below, isstraightforward.

Lemma 4. If c�1ðmÞ or f0�1ð0Þ contain no other elements, then the index functions of

ðd3gÞ � ðb3aÞ and ðd � bÞ3ðg � aÞ are the same and middle four interchange holds for the

fences themselves.

If the conditions of Lemma 4 are not satisfied, cðl � 1Þ ¼ m and f0ð1Þ ¼ 0; andthe relevant part of the pasting diagram is as follows:

ð17Þ

We can factorize a as

a ¼ * � a; ð18Þ

where a is obtained from a by the substitution

while * ¼ 13iðq1;y;qm;p0;y;pmÞ; i.e.,


Analogously, we can factorize d as

d ¼ d � *Z; ð19Þ

where d is obtained from d by the substitution

and

In the rest of this proof we use the following notation for horizontal and verticalcomposition of 2-cells:

(blocks of 2-cells separated by lines should be composed first, as if parenthesized).We first prove that the following instance of middle-four interchange holds:

ð20Þ

This is the pasting of the following fences:

ð21Þ


Performing the horizontal compositions first gives:

and then by vertical composition, we obtain

ð22Þ

Composing the fences in diagram (21) in the vertical direction first gives

which after horizontal composition yields

ð23Þ

Fences (22) and (23) are directly equivalent via the identity arrow labelledI in (23).


We are now able to prove the general case of the middle-four interchange law:

completing the proof of the theorem. &

The rather brief appearance of the equivalence relation in the previous proofmight lead us to wonder whether its full strength is needed; the next theorem showsthat it is.

Theorem 2. The equivalence relation given in Definitions 1 and 2 is the minimal

relation such that horizontal and vertical composition of equivalence classes of fences

satisfies the middle-four interchange law.

Proof. We need to show that if any two fences are directly equivalent, there is aninstance of the middle-four interchange law which implies that they should be equal.Because of the induction argument in the proof of the previous proposition, it issufficient to check the following two cases of the equivalence relation.


In case 1, y is of the form

and o is of the form

In case 2, y is of the form

and o is of the form

We will do case 1 and leave case 2 to the reader. The algorithm of the proof ofProposition 1 decomposes y as a pasting of the following fences:


and o can be decomposed as

Since hj ¼ lgi and ki�1 ¼ qjl; these decompositions can be rewritten in the following

way. y is the pasting of the fences


and o is the pasting of the fences

It is now obvious that one can be obtained from the other by middle-fourinterchange. &

4. Universal property of P2C

There is an obvious embedding functor ð�Þ*:C-P2C; which is the identity on

objects and sends morphisms to sequences of length 1: f*¼ ðf Þ: It is easy to check

that it has the following property:

Lemma 5. The embedding ð�Þ*:C-P2C is locally fully faithful.

It also has the property that it sends arrows in C to arrows which are left adjointsin P2C: We will call any functor which sends arrows to left adjoints sinister.We will also need the notion of a sinister natural transformation between sinister

functors. Let F ;G :C4D be sinister functors, and let a : F ) G be a strongtransformation. Recall (cf. [1]) that this means that a is determined by a family ofarrows aA : FA-GA in D; for objects AAC0 together with a natural family ofisomorphisms af : aBFf ) Gf aA in D; for arrows f :A-B in C: For any arrow

f :A-B; choose right adjoints ðFf Þn and ðGf Þn for Ff and Gf ; respectively.Together with af these give rise to the following squares in D:


where bf : ðGf ÞnaB ) aAðFf Þn is the mate of a�1f ; i.e.,

bf ¼ ððGf ÞnaB Ff Þ � ððGf Þna�1f ðFf ÞnÞ � ðZGf aAðFf ÞnÞ:

Definition 3. A strong natural transformation a : F ) G between two sinisterfunctors F ;G :C4D is called sinister if aIA

¼ iaAfor all objects AAC0: Moreover, it

is called strongly sinister if for each arrow fAC1 the mate bf of a�1f is an

isomorphism.

Recall (cf. [1]) that in general a natural transformation a between two bifunctorsF ;G :C4D can be represented by a bifunctor Ka :C-CylðDÞ: This bifunctor isdefined as follows: on objects,

on arrows,

and on 2-cells, Kaðif Þ is the cylinder with identity 2-cells on the top and the

bottom and af on the front and the back. Definition 3 is justified by the following

result.

Proposition 2. A transformation a : F ) G between sinister 2-functors

F ;G :C4D is sinister if and only if its representing bifunctor Ka :C-CylðDÞ is a

sinister 2-functor.

Proof. Let a : F ) G be a sinister (strong) natural transformation between sinister2-functors, and let Ka :C-CylðDÞ be its representing functor. Let fAC1 be any

morphism. We claim that bf ; the mate of a�1f as defined above, is right adjoint to

Kaf : The unit and counit of the adjunction are the pairs ðZFf ; ZGf Þ and ð Ff ; Gf Þrespectively. Moreover, Ka is a 2-functor, since KaðIAÞ ¼ iaAIFA

for all AAC0 and itpreserves composition.To prove the converse, suppose that a : F ) G is a natural transformation

between sinister 2-functors, such that its representing bifunctor Ka :C-CylðDÞ is asinister 2-functor. Since Ka is a 2-functor, we have that aIA

¼ iaA: To check that a is a

strong transformation, let fAC1 be any morphism. Since Ka is sinister, there exists an


arrow ðKaf Þn in CylðDÞ; i.e., a square

where ðFf Þn and ðGf Þn are right adjoint to Ff and Gf ; respectively. The mate of

ðKaf Þn can be shown to be the inverse of af : &

Let WeakðC;DÞ denote the category of 2-functors and weak natural transforma-tions, let StrongðC;DÞ denote the category of 2-functors and strong naturaltransformations, let SinðC;DÞ denote the category of sinister 2-functors and sinisternatural transformations, and let SSinðC;DÞ denote the category of sinister 2-functors and strongly sinister natural transformations.

Theorem 3. The functor ð�Þ*:C-P2C has the property that for each arrow f in C

the arrow f*

in P2C has a right adjoint in P2C (i.e. f*

is a left adjoint). Moreover, this

functor is universal with this property in the sense that composition with ð�Þ*

defines

the following equivalences of categories

WeakðP2C;DÞ -B SinðC;DÞ;

StrongðP2C;DÞ -B SSinðC;DÞ:

Proof. Let F :C-D be a sinister 2-functor. Our first goal is to construct a 2-functor

F :P2C-D such that F3ð�Þ*DF : Choose, for each arrow ðf :A-BÞAC1; a specific

arrow ðFf ÞnAD1; and 2-cells

ZFf : IFðAÞ ) ðFf ÞnFðf Þ and Ff : Fðf ÞðFf Þn ) IFðBÞ

such that ZFf and Ff satisfy the triangle identities, i.e.,

Fðf Þ Ff � ZFf Fðf Þ ¼ iFðf Þ and Ff ðFf Þn � ðFf ÞnZFf ¼ iðFf Þn :

On objects, F is the same as F : On arrows, F is defined by

Fðg1;y; gn; f0;y; fnÞ ¼ FðfnÞðFgnÞnFðfn�1Þ?ðFg1ÞnFðf0Þ:

Since the 2-cells in P2C are generated, by identities, ’s, and Z’s as in (4),

it sufficient to define F for these particular 2-cells, which is done in the


obvious way:

The functor F is well defined on equivalence classes of 2-cells, since thisequivalence relation is equivalent to having middle-four interchange for the verticaland horizontal composition of fences (cf. Theorem 2). Finally, it is clear that

F3ð�Þ*¼ F : This shows that the functor ‘composition with ð�Þ

*’ is essentially

surjective on objects for both equivalences described in the theorem.It remains to be shown that for any two functors F ;G :P2C4D; the functor

‘composition with ð�Þ*’ induces an isomorphism between the appropriate sets of

natural transformations, i.e.,

WeakðF ;GÞDSinðF3ð�Þ*;G3ð�Þ

*Þ and StrongðF ;GÞDSSinðF3ð�Þ

*;G3ð�Þ

*Þ:

Let a : F3ð�Þ*) G3ð�Þ

*be a sinister natural transformation. By Proposition 2, its

representing functor Ka :C-CylðDÞ is sinister. We need to show that there is aunique natural transformation *a : F ) G; such that *a3ð�Þ

*¼ a: To obtain K*a; take

the lifting Ka :P2C-CylðDÞ of Ka such that Kaðf Þ is the square

where bf is the mate of a�1f : It is clear that this is the only lifting of Ka for which the

top is F and the bottom is G: Moreover, a is strongly sinister if and only if *a is astrong transformation. &


4.1. Adding right adjoints for a class of arrows

As mentioned in the introduction, the Span construction adds adjoints freelysubject to the Beck condition. If instead one adds adjoints only for monomorphisms(still subject to the Beck condition), one obtains the closely related partial mapconstruction. Similarly, in the calculus of fractions one usually obtains more usefulresults by inverting only a distinguished class of arrows (e.g., the homotopyequivalences) in a category.In the same way, we may want to add right adjoints for only a class of arrows

WCC: To do this it is necessary to require that the gi in every zig-zag (6) be in W :The class of fences with such domains and codomains is closed under horizontal andvertical composition. The proofs that horizontal and vertical composition are well-defined (Lemmas 2 and 3) and of the middle-four interchange property (Theorem 2)do not involve the creation of new backward arrows; thus these fences form a

2-category, which we will denote by C½W n :Since the proof of the universal property of P2C (cf. Theorem 3) does not involve

the creation of any new backward arrows either, we have in fact proved a universal

property for C½W n : Define W -sinister 2-functors and (strongly) W -sinister naturaltransformations in the obvious way. Let SinW ðC;DÞ denote the category of W -sinister 2-functors and W -sinister natural transformations; and let SSinW ðC;DÞdenote the category of W -sinister 2-functors and (strongly) W -sinister naturaltransformations.

Theorem 4. Let C be a category and WCC1 a distinguished class of arrows. The

functor ð�Þ*:C-C½W n has the property that for each arrow f in W the arrow f

*in

C½W n has a right adjoint in C½W n (i.e., f*

is a left adjoint). Moreover, this functor is

universal with this property in the sense that composition with ð�Þ*

defines the

following equivalences of categories:

WeakðC½W n ;DÞ -B SinW ðC;DÞ;

StrongðC½W n ;DÞ -B SSinW ðC;DÞ:

5. Some basic properties

5.1. Arrows with right adjoints

We have shown that the arrows of the form f*¼ ðf Þ in P2C have right adjoints.

For a category C with pullbacks, we know that in SpanC the only left adjoints are


the arrows in the image of C: Although P2C and SpanC are distinct categories ingeneral, we see that this result also holds for P2C; we show that up to 2-isomorphism, the arrows of the form f

*are the only left adjoints in P2C: Using the

notation of [2], MapðP2CÞDC:

Proposition 3. Let C be any category. If a morphism ðg1;y; gn; f0;y; fnÞ in P2C has

a right adjoint, there exists an arrow fAC1; with a 2-isomorphism

ðg1;y; gn; f0;y; fnÞDðf Þ:

Proof. Suppose that the arrow ðg1;y; gm; f0;y; fmÞ :A-B has a right adjoint inP2C; say ðq1;y; qn; p0;y; pnÞ :B-A: Let the unit and counit of the adjunction berepresented by the fences

and

respectively. One of the triangle identities states that the following composition of 2-cells is the identity 2-cell for the morphism ðg1; f0; f1Þ: (We draw the diagram only form ¼ 1:)


Vertical composition results in

ð24Þ

Since this 2-cell is equivalent to the identity 2-cell, there is an arrow l : B0-Am;such that wm ¼ lf0 and fml ¼ vnþ1:We claim that ðg1;y; gm; f0;y; fmÞ is isomorphicto ðfmlf0Þ in P2C: Indeed, the required 2-cells are

and

It is easy to see that Y1Y2 ¼ idfmlf1 (after all, the original category has only identity

2-cells). The composite Y2Y1 is equal to the cell represented by fence (24), and byassumption this cell is equivalent to the identity 2-cell. &

5.2. Limits and colimits in P2C

The non-empty hom-sets (we are considering the underlying category of P2ðCÞ)are always infinite; if A-

f0’g1

y-fn

B is a zig-zag, then so is A-f0’g1

y-fn’fn-fn

B;

and they are not equal. Thus, it follows that except in vacuous cases, P2ðCÞ neverhas a terminal (or initial) object.Similar arguments show that the underlying category of P2ðCÞ has only trivial

products.


Proposition 4. The product A � B exists in P2ðCÞ if and only if A ¼ B and no arrow

except for IA maps into A; in this case, A � A ¼ A:

Proof. Suppose that A � B exists in P2ðCÞ; then the projections

A’p0

A � B-p1

B

are zig-zags

A’fm

y-g1

’f0

A � B-f 00’g01y-

f 0nB

and it is evident that the zig-zags

A’fm

-fm

’fm

y-g1

’f0

A � B-f 00’g01y-

f 0n’f 0n-f 0n

B

can only factor through the projections if

f0 ¼ g1 ¼ f1 ¼ ? ¼ fm ¼ f 00 ¼ g0

1 ¼ f 01 ¼ ? ¼ f 0

n: ð25Þ

As we always have the option of using an identity within a zig-zag, (25) is only forcedunder the conditions stated in the proposition. &

Note. The dual of this proposition also holds; the only coproducts in P2ðCÞ are ofthe form A

‘A where only IA maps out of A:

However, there may be nontrivial equalizers and coequalizers, and there arealways nontrivial pullbacks and pushouts.

Proposition 5. A diagram A4f0

f 00

5g1

g01

?4fm

f 0n

B has an equalizer in P2ðCÞ if and only if

m ¼ n; g1 ¼ g01;y; fm ¼ f 0

m; and A4f0

f 00

X has an equalizer in C:

Proof. If the two zig-zags A-f0’g1

?-fm

B and A-f 00’g01 ?-

f 0n

B are different lengths,

or if they differ anywhere except at the extreme left, we can never have

ðA0 -f 000’g001 ?-

f0f00

k’g1

?-fm

BÞ ¼ ðA0 -f 000’g001 ?-

f 00f 00k’g01 ?-

f 0nBÞ: ð26Þ

Conversely, if m ¼ n; g1 ¼ g01;y; fm ¼ f 0

m; the zig-zags in (26) are equal if and only if

f0f00

k ¼ f 00f

00k : Should this be the case, the length-1 zig-zag A0 -

f 00k

A satisfies (26) itself,

and cannot factor through any longer zig-zag; so only length-1 zig-zags can be


equalizers in P2ðCÞ: Finally, if

E -e

A4f0

f 00

B

is an equalizer diagram in C; then f0e ¼ f 00e; and if (26) holds, A0 -

f 000’g001 ?-

f 00k

B

factors uniquely up to isomorphism through e; as A0 -f 000’g001 ?-

qE -

eA where

eq ¼ f 00k in C: &

We thus see that the existence of non-trivial equalizers in P2ðCÞ depends on theirexistence in C: Coequalizers are, of course, computed analogously. The nextproposition, too, has two dual forms for pushouts and pullbacks; in this case, thecolimit form is notationally easier.

Proposition 6. A diagram

A’fm

y-g1

’f0

C -f 00’g01y-

f 0n

B ð27Þ

has a pushout in P2ðCÞ if either

ðiÞ f0 ¼ f 00; fi ¼ f 0

i and gi ¼ g0i for i ¼ 1;y;minðm; nÞ; or

ðiiÞ m ¼ n; f0 ¼ f 00; fi ¼ f 0

i and gi ¼ g0i for i ¼ 1;y;m � 1; gm ¼ g0

m; and A’fm

X -f 0m

B

has a pushout in C:

Proof. We may suppose, without loss of generality, that mpn: We consider twocases, depending upon whether mon or m ¼ n: In the first case, if we can fill out thesquare at all, there exists

A-fm

’gmþ1

y-fk

D’f 0ky-

g0nþ1

’f0n

B ð28Þ

such that

f0 ¼ f 00; g1 ¼ g0

1;y; fmfm ¼ f 0m;y; fn ¼ f0nf 0

n;y; fk ¼ f 0k: ð29Þ

A necessary condition on (27) for this to happen is that

f0 ¼ f 00; g1 ¼ g0

1;y; gm ¼ g0m


and that fmAC1 with fmfm ¼ f 0m should exist. We can fill the square out with

A-fm

’gmþ1

y-fn

B’IB

B: ð30Þ

Any other commuting completion (28) factors uniquely (up to isomorphism)

through B-fn

’gnþ1

y-fk

D: We thus conclude that (30) is a pushout. We note that,

while fm may not be unique, all choices give isomorphic pushouts.If, on the other hand, m ¼ n; then we have, instead of (29),

f0 ¼ f 00; g1 ¼ g0

1;y; fmfm ¼ f0mf 0m;y; fk ¼ f 0

k ð31Þ

and (27) takes the form

A’fm

X -gm

y-g1

’f0

C -f0’g1

y’gm

X -f 0m

B: ð32Þ

Again, any pair fm; f0m consistent with (31) allows us to close the square with

A-fm

Y ’f0m

B: ð33Þ

If (33) is the pushout of

A’fm

X -f 0m

B ð34Þ

in C; any commuting completion

A-fm

Y ’gmþ1

y-fk

D’fk

y -gmþ1

Y’f0m

B

of (32) can be factored (in P2ðCÞ) through (33) by the zig-zag

Y -h

Y ’gmþ1

y-fk

D;

where h is the unique arrow factoring A-fm

Y’f0m

B through (33) in C: Again, we

conclude that (33) is a pushout in P2ðCÞ: Conversely, if (34) does not have a pushoutin C; it does not have one in P2ðCÞ: &

We have, then, obtained necessary and sufficient conditions for the existence ofspecific pushouts in P2ðCÞ: These conditions are such that we can see that (on theone hand) P2ðCÞ never has all pushouts except in trivial cases; but (on the otherhand) it always has some pushouts. Clearly, analogous conditions govern theexistence of pullbacks.


The virtual non-existence of products, and the restrictions on limits of other types,are not peculiar to P2ðCÞ: Rather, they result from the formal nature of thiscategory, which means that only rather predictable factorizations exist. The freecategory on a graph has similarly restricted limits.It may be noted that the conditions of this section (especially the non-existence of

non-trivial products) give a quick way to show that most familiar 2-categories do notarise as P2ðCÞ for any C:

6. P2 of a free category

Let G be a directed graph, and FG the free category generated by G: Allmorphisms are mono and epi, so the word problem for 2-cells in P2FG is solvable.In fact there is a nice geometric description of the 2-cells.As the morphisms of FG are paths of edges of G all in the forward direction,

the morphisms of P2FG will be paths of edges of G going in eitherdirection:

A0-a1

A1-a2

A2’a3

A3-a4

A4’a5

A5’a6

A6:

We denote such a path by ðaþ1 ; aþ

2 ; a�3 ; aþ

4 ; a�5 ; a�

6 Þ; where the superscript denotesthe direction of the edge.Strictly speaking, the morphisms of P2FG are a bit more complicated than this as

we are allowing our paths to contain identities. This makes the general theory easierand is necessary for the two end morphisms. As identities are paths of length zero inFG; it is impractical to put them in here. This does not affect the universal propertyas we get a locally equivalent 2-category with the same objects. Indeed if in thegeneral construction a path of length k contains

?Ai -fi1

Bi ’1Bi

Bi -fiþ1

Biþ1?

then the cells


are inverse isomorphisms with the path of length k � 2 in which the identity isremoved and the adjacent arrows composed.

Let a ¼ ða ðiÞi Þipp and b ¼ ðbZðjÞ

j Þjpq be two paths with common domain and

codomain. In order to describe the 2-cells a ) b; we recall the following definitionfrom [9]. A ðp; qÞ diagram is a partition P of the set f1; 2;y; p þ qg into subsets ofsize two. We represent this pictorially as a rectangle with p marked points on the topedge numbered from 1 to p and q on the bottom numbered from p þ 1 to p þ q: Iffi; jgAP; the point i is joined to the point j:

It is a planar ðp; qÞ-diagram if corresponding points can be joined by non-intersecting arcs which lie inside the rectangle, as in the above example. (It would notbe planar if, for example, f4; 6g and f5; 7gAP:) Planar ðp; qÞ-diagrams are also calledKauffman diagrams in [10], at least when p ¼ q: They form a special case of the affinediagrams of Graham and Lehrer [8].

Given paths a and b as above, define aðiÞ

i ¼ bZði�pÞi�p for poipp þ q: We define a

planar ða; bÞ-diagram to be a planar ðp; qÞ-diagram such that for all fi; jg in P wehave

ai ¼ aj

and

ðiÞ ¼ � and ðjÞ ¼ þ if iojpp;

ðiÞ ¼ þ and ðjÞ ¼ � if poioj;

ðiÞ ¼ ðjÞ otherwise:

Thus the points that are joined are labeled by the same edge. These edges go in thesame direction if one is on top and one on the bottom, and in opposite directionsotherwise. If two top points are joined the edges point away from each other whereason the bottom they point toward each other. These conditions on the directions ofjoined edges can be coded into an orientation on the joining strands. They areoriented so that rotating from the forward direction of the edge to the forward


direction of the string is clockwise:

The strands joining points at the same level go from right to left.Planar ða; bÞ-diagrams can be pasted horizontally or, if they are compatible,

vertically. Because of the orientations of the strands no loops are formed whenpasting.

Proposition 7. Given two morphisms a and b in P2FG as above, there is a bijection

between 2-cells a ) b and isotopy classes of planar ða; bÞ-diagrams. Furthermore,horizontal (vertical) composition of cells corresponds to horizontal (resp. vertical)pasting of diagrams.

Proof. Suppose we have a morphism in P2FG represented by a diagram such as thefollowing:

Each commutative square in this diagram is an equality of arrows in the freecategory FG; so the same edges appear in each composite and in the same order. Thecorresponding edges may be joined by non-intersecting curves with the correctorientation. Each edge occurring in a vertical arrow borders two regions so there willbe a curve entering and one leaving. Thus the end points of the pasted curves lie onthe top or bottom of the diagram, giving a planar ða; bÞ-diagram. As the verticalarrows consist of downward edges, the strands cross them from right to left.The ða; bÞ-diagram structure is independent of which representative of the 2-cell

we choose. Let l be as in the diagram below and consider the two squares adjoiningthe one containing l (there are four possible combinations of shapes and they allwork the same).


Our equivalence says that we can replace this diagram by one of the form

The curves joining corresponding arrows in the middle square can be drawn by firstjoining corresponding edges in each of the triangles made by l and then pasting thosethat pass through l together. These curves are now pasted to the matching ones in thesquares on the right and left. But this could have been accomplished by erasing the h

and k (resp. h0 and k0) but keeping the l and then joining the corresponding edges inthe remaining two commutative pentagons. The result will be the same in both cases.To show that every planar ða; bÞ-diagram represents a 2-cell, note first of all that if

every top edge is paired with a bottom edge and conversely, then the diagramrepresents the identity 2-cell. Otherwise, there will be two top (or bottom) edgespaired together and so two adjacent ones paired together. Remove these two and weget a smaller ða; bÞ-diagram which, by induction, represents a 2-cell. The originaldiagram is obtained by pasting this diagram with one having just two adjacent edgesjoined and augmented with identities on either side. For example

can be written as

The top diagram represents the 2-cell


with the appropriate identities pasted on either side, i.e.

&

The hom categories in P2FG have a lot of nice structure. For example they haveepi-mono factorizations.

Proposition 8. An ða; bÞ-diagram is an epimorphism (monomorphism) if and only if

there are no bottom (resp. top) hoops.

Proof. Consider an ða; bÞ-diagram a with a hoop on the bottom as follows:

If we compose with

we get the same result, namely a diagram like a but with two hoops instead of one.So a is not epi.Conversely, if no bottom vertex is joined to another bottom vertex, then

each bottom vertex is joined to a top one with the same label and the same sign.If we paste a with a cell b; we can recover b from a � b by picking out thosethrough strands of a together with the strands they are joined to in b:


For example

which is isotopic to b: Thus a is epi.The corresponding statement for monos is the same. &

The reader may compare Lemma 1.6 of Graham and Lehrer [8].Now given an ða; bÞ-diagram, it can be factored into an epi followed by a mono as

follows. Deform the strands isotopically so that no hoop is more than one third theheight of the diagram. Then draw a line across the middle of the diagram and it isfactored.

The description of monos also follows by the duality P2FðGÞcoDP2FðGopÞobtained by taking an ða; bÞ-diagram, changing the sign of each node and thenreflecting in a horizontal line. For example,


There is another duality, namely P2f ðGÞopDP2f ðGÞ; obtained by reflection in avertical line and changing the signs of the nodes:

Let us clarify what is happening with dualities. For an arbitrary category A we

always have P2ðAopÞDP2ðAÞcoop obtained by reversing everything in our diagramsfor 2-cells. But there is also another duality which holds for an arbitrary category,

namely ð Þn: P2ðAÞopCP2ðAÞ which is the identity on objects. An arrow, which is azig-zag path of morphisms from A; is sent to the same zig-zag in the reverse directionwith identities added at each end to give it the right shape and the same is done to 2-

cells. ð Þnn is not exactly the identity. It is on objects but only induces an equivalenceon the hom categories. We might call it a pseudo-equivalence. This is not a seriousproblem and arises only because of our choice of shape of zig-zag (starting andending with forward arrows). If we combine these two equivalences we get

P2ðAopÞCP2ðAÞco: As we are ignoring identities in the case of free categories, thisgives an actual equivalence P2FðGÞcoDP2FðGopÞ:We now consider some special cases.

Example 2. The simplest non-trivial example is for the graph 2 ¼ 0-1: A path from0 to 0 must look like

0-1’0-1’0-1’0

i.e. a number nAN of 0-1’0: An ða; bÞ-diagram from ða�aþÞn to

ða�aþÞm must look like

i.e. there are no nested hoops. Furthermore, the region between a down strand and

the up strand immediately to its right must involve an aþ on the bottom and the a�

immediately to its right otherwise the a� to its right could be joined to nothing. Thusto each aþ; a� on the top there is associated a unique such pair on the bottom and


planarity says that this is an order preserving map n-m: Conversely, any orderpreserving map n-m is realized this way. Indeed if j : n-m is order preserving,

then we join the ith aþ on the top to the jðiÞth aþ on the bottom and the jðiÞth a�

on the bottom to the jth a� on the top where j is the largest number such thatjðjÞ ¼ jðiÞ: Those not thus joined are joined to the appropriate neighbor. Thus wesee that ða; bÞ-diagrams from ða�aþÞn to ða�aþÞn are in bijection with orderpreserving maps n-m: So the hom category P2Fð2Þð0; 0Þ is D; the category of finiteordinals.Paths 0-1 look like

0-1’0-1’0-1’0-1

i.e. aþða�aþÞn: An ða; bÞ-diagram from aþða�aþÞn to aþða�aþÞm might look like

But now the last aþ on the bottom cannot be paired with anything else on the

bottom so must be paired with an aþ on the top. So with the same argument as

above applied to the aþs, we see that such a diagram corresponds to an orderpreserving map nþ 1-mþ 1 which preserves the top element. Thus the homcategory P2Fð2Þð0; 1Þ is D

*; the category of non-zero finite ordinals and top

preserving morphisms.

By the duality P2FðGÞcoDP2FðGopÞ mentioned above, we see that the homcategory P2Fð2Þð1; 1Þ is Dop and P2Fð2Þð1; 0Þ is Dop

*: Hence we recover the

Schanuel–Street description of the free adjoint [12].

Example 3. A related example is for the graph G with 2 vertices, 0 and 1, and a set C

of edges 0-1 (colors). Paths from 0 to 0 will be as in Example 2 but now each nodeis assigned a color. The 2-cells are ða; bÞ-diagrams but now the threads are alsocolored, joining nodes of the same color.

7. Further research

In the previous section we have seen some examples of the surprisingly richstructure of 2-categories which arise as P2 of a free category. Further results in thisarea and more examples will be discussed in a forthcoming paper.The ða; bÞ-diagrams discussed above resemble Kauffman diagrams, but have

different properties due to their directedness. If instead one freely adds arrows to a


free category which are simultaneously left and right adjoint to the existing arrows,the 2-cells can be described as ‘honest’ Kauffman diagrams without directions. Thisconstruction together with some applications will also be discussed in a forthcomingpaper. There we will also consider presentation for 2-cells of P2 of an arbitrarycategory in terms of equivalence classes of diagrams. In the general case, the arcs joinand branch (representing composition and factorization), and thus resembleFeynman diagrams rather than Kauffman diagrams.

References

[1] J. Benabou, Introduction to bicategories, in: Reports of the Midwest Category Seminar, Lecture

Notes in Mathematics, Vol. 40, Springer, New York, 1967, pp. 1–77.

[2] A. Carboni, R.F.C. Walters, Cartesian bicategories I, JPAA 49 (1987) 11–32.

[3] R.J. MacG. Dawson, R. Pare, D.A. Pronk, Undecidability and free adjoints, in: N. Callaos, F.G.

Tinetti, J.M. Champarnaud, J.K. Lee (Eds.), Proceedings of the World Multiconference on Systemics,

Cybernetics and Informatics 2001, Vol. XIV, International Institute of Informatics and Systemics,

Orlando, 2001, pp. 156–161.

[4] R.J. MacG. Dawson, R. Pare, D.A. Pronk, Undecidability of the free adjoint construction, preprint.

[5] R.J. MacG. Dawson, R. Pare, D.A. Pronk, Spans and free adjoints, to appear.

[6] W.G. Dwyer, D.M. Kan, Calculating simplicial localizations, JPAA 18 (1980) 17–35.

[7] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, New York, 1967.

[8] J.J. Graham, G.I. Lehrer, The representation theory of affine Temperley–Lieb algebras, L’Enseign.

Math. (2) 44 (3–4) (1998) 173–218.

[9] V.F.R. Jones, A quotient of the affine Hecke algebra in the Brauer algebra, L’Enseign. Math. 40

(1994) 313–334.

[10] Z.A. Landau, Fuss–Catalan algebras and chains of intermediate subfactors, Pacific J. Math. 197 (2)

(2001).

[11] D.A. Pronk, Etendues and stacks as bicategories of fractions, Comput. Math. 102 (1996) 243–303.

[12] S. Schanuel, R. Street, The free adjunction, Cahiers Topologie Geom. Differentielle Categoriques 27

(1986) 81–83.


Documents

Adjoining adjoints