Whitney 1-categories and Whitney functors, i.e. natural transformations, be- tween them form the category 1W hit. Dagger categories and dagger functors be- tween them form a categoryDag. In factDagis a 2-category but we will restrict our attention, at least in this thesis, to the 0-morphisms and 1-morphisms.
Theorem 8.8. We have functors
1W hit C ' ' Dag. W f f
which are inverse to each other and so have an equivalence of categories, i.e. given a Whitney functor η:A ÑA1 of Whitney 1-categories there is a functor
8.3 The Relationship Between the Constructions
Cpηq:CpAq ÑCpA1qbetween the corresponding dagger categories. And in the opposite direction, given a functorF :C ÑD between dagger categories, there is a natural transformationWpFq:WpCq ÑWpDq between the corresponding Whitney 1-categories.
Proof. Take a Whitney functor, i.e. natural transformation of Whitney 1- categories η:A ÑA1. This tells us that given an object X PP restrat1 there exists a morphismηX :ApXq ÑA1pXqinSetssuch that for every prestratified
mapα:X ÑY PP restrat1 the square ApXqooApαq ηX ApYq ηY A1pXqooA1pαq A1pYq
commutes. To define a functor we must have a mapObjCpAq ÑObjCpA1q. The
natural transformation gives us such a map, ηpt : Apptq Ñ A1pptq. Similarly
there should be a mapping from the morphisms in CpAq to those in CpA1q, the obvious candidate for this is ηI : ApIq Ñ ApIq. Taking the above square
and choosingX I and Y I2, and setting Apαq c and A1pαq c1, each an example of the composition map previously defined, we get the commuting square, ApIqoo c ηI ApI2q ηI2 A1pIqoo c1 A1pI 2q From this we see that
ηIpcpf, gqq c1pηI2pf, gqq, i.e.
ηIpf gq c1pηIpfq, ηIpgqq ηIpfqηIpgq,
and so composition is preserved. HereηI2 ηIηI.Similarly consider
ApIqoo id ηI Apptq ηpt Apptqoo s ηpt ApIq ηI A1pIqoo id1 A1p ptq A1pptqoo s1 A1pIq
These commuting squares show
and spηptpfqq ηIpspfqq
and similarly for t. Hence we have constructed a functor Cpηq:CpAq ÑCpA1q.
What about the other direction? Given a functor F :C Ñ D between dagger categories can we define a functor WpCq Ñ WpDq between the corresponding Whitney 1-categories? We need a natural transformation assigning to eachX P Strat1, a map of setsWpCqpXq ÑWpDqpXq,mapping the equivalence classes of labellings ofXinduced byC,to equivalence classes of labellings induced byD. The fact that we have a functorC ÑDmeans we have maps, i.e. ObjC ÑObjD and M orC ÑM orD. Consider the following diagram:
WpCOOqpXq WpFqpXq // f WpDqpOO Xq f WpCqpYq WpFqpYq / /WpDqpYq
We defineWpFqpXq by applyingF to all labels ofX by C to obtain a labelling byD.This square commutes becauseF is a functor and so preserves composition and identities. Therefore WpCqpFq is a natural transformation.
The constructions
C: 1W hitÑDag W :Dag Ñ1W hit
are inverse to each other in the sense that there is a natural isomorphism of dagger categories
C ÑCpWpCqq, and a natural isomorphism of Whitney 1-categories
A ÑWpCpAqq.
To see this fix a dagger category C, from it construct a Whitney 1-category WpCqand then use this to define a new dagger category CpWpCqq.As we have previously defined, to construct a dagger category from a Whitney 1-category we assign WpCqpptq ObjCpWpCqq.Now, an element ofWpCqpptq is a labelling
of the point, we use the objects ofC to label vertices and soWpCqpptq ObjC. And so
8.3 The Relationship Between the Constructions
Similarly we get the bijection,
M orCpWpCqqWpIq tlabellings of an edgeu{M orC.
This identification respects sources and targets (by the definition of labelling) and composition. Thus we have defined a dagger functor
C Ñ CpWpCqq, (8.1)
which is an isomorphism of categories. To see this isomorphism is natural con- sider the commuting square
C F // C1 CpWpCqq CpWpFq //CpWpC 1qq.
We can either apply the functorF toC and then constructCpWpC1qor equiv- alently construct CpWpCq and then apply the functor to the labels to get CpWpC1q,i.e the isomorphism is natural.
Next we need to define a functor between Whitney 1-categories, i.e. a natural transformation
A ÑWpCpAqq. (8.2)
We do this by compatibly defining,
ApXq ÑWpCpAqqpXq (8.3) for allX PStrat1. HereCpAq is the dagger category constructed from A and WpCpAqq is the functor which assigns an equivalence class of labellings from CpAq to eachX PStrat1.
First choose an orientation of the edges ofX. GivenaPApXq, we want to assign an element ofApptq to each vertex and an element of ApIq to each (oriented)
edge in a compatible way. The characteristic map for a vertex, pt ÝÑu X, gives ApXqÝÑu Apptq. We therefore label this vertex by ua.
Since we have choosen an orientation there is a characteristic map,IÝÑv X, onto
an oriented edge, givingApXqÝÑv ApIq and so we label this edge byva. This
gives a labelling ofX by CpAq,i.e. an element ofWpCpAqqpXq. AsA is a sheaf, we have
ApXq lim
It follows from this that p8.3q is a bijection. To see that (8.2) is a natural isomorphism fix an objectX and consider the commuting square,
ApXq ηX // A1pXq WpCpAqqpXq WpCpηXqq / /WpCpA1qqpXq.
Given a P ApXq we apply ηX to get an element in A1pXq and then apply
the construction to get an element of WpCpA1qqpXq. Equivalently, given a P ApXq we first apply the construction to get an element of WpCpAqqpXq, i.e. an equivalence class of labellings of X by CpAq, and then apply ηX to all the
labels to get an equivalence class of labellings of X by elements of CpA1q,i.e. an element of WpCpA1qqpXq.
Chapter 9
One-Object Whitney
2-Categories
We now direct our attention to one-object Whitney 2-categories. Recall a Whit- ney 2-category is a presheaf A : P restrat2 Ñ Sets which is a sheaf when restricted toStrat2. In this section we assume there to be only one object, i.e. Apptq 1. We will give a construction similar to the Whitney 1-category case which from a one-object Whitney 2-category produces a dagger pivotal cate- gory. This is the notion of a one-object 2-category with duals that we adopt. In the other direction we will give constructions that produce one-object Whitney 2-categories from dagger pivotal categories.
9.1
Constructing a Dagger Pivotal Category from a
One-Object Whitney
2-Category
In this section we will construct a functor between the category of dagger pivotal categories and the category of one-object Whitney 2-categories,
C: 2W hit1 ÑDagP iv.
Definition 9.1. FromA we define a category CpAq. We set ObjCpAqApIq,
and
where α|tI Apfqpαq forf :ttu I ãÑI2.We also use 1PApIq for image of
the the unique element in Apptq,induced from the unique map of I to a point.
We have four maps:
• The source and target maps, s,t : ApI2q Ñ ApIq, are induced from the
inclusions i0 : I Ñ I2 where t ÞÑ pt,0q for all t and i1 : I Ñ I2 where
tÞÑ pt,1q for all t, respectively.
• The mapid:ApIq ÑApIIqis induced from the projection ofI2onto its
first factor. This map takes an object and returns the identity morphism on that object.
• The composition mapc:ApI2q ApIqApI
2q ÑAp
I2qis induced from the
mapI2ÑII2 :px, yq ÞÑ px,2yq.Recall that the sheaf property tells us ApII2q ApI2q ApIqApI
2q.
Lemma 9.2. The maps in Definition 9.1 satisfy:
• Source and target of composites: spcpf, gqq spfq and tpcpf, gqq tpgq. • Associativity of composition: cpcpf, gq, hq cpf,cpg, hqq.
• Identities act as units for composition: cpida, fq f cpf,idbq, where
pf :aÑbq PM orCpAq.
Proof. The proof of this lemma is the same as that for Lemma 8.3 in the Whitney 1-category pn1q case, but here we carry along an extra factor. For example, the equality in the second property comes from the homotopy
Hpx, y, tq "
px,2y 2tyq ify P r0,12s px,4y 2tp1yq 1q ify P r12,1s which should be compared with the homotopy H in Lemma 8.3.
Remark 9.3. Since we have defined Apptq 1 we have ApI2q ApIq ApptqApIq
ApIq ApIq
Lemma 9.4. The categoryCpAq has a specific monoidal structure.
Proof. We define a bifunctorb:CpAqCpAq ÑCpAq.On objects this functor is the map
b:ApIq ApIq ÑApIq:pA, Bq ÞÑAbB
9.1 Dagger Pivotal Cat from a One-Object Whitney 2-Cat
On morphisms it is the map
b:ApI2q ApIqApI
2q ÑAp
I2q:pf, gq ÞÑf bg
induced by the prestratified mapI2 ÑI2I:px, yq ÞÑ p2x, yq, see Figure 9.1.
Figure 9.1: The mappx, yq ÞÑ p2x, yq,inducing the tensor product onCpAq
The fact that the functor respects composition is illustrated in Figure??. More explicitly composing then tensoring is induced from the composite
px, yq ÞÑ px,2yq ÞÑ p2x,2yq
and tensoring then composing is induced from the composite px, yq ÞÑ p2x, yq ÞÑ p2x,2yq.
These are different factorisations of the same map and so induce the same map of sets, giving the ‘exchange law’
pfbgq pf1bg1q pff1q b pgg1q.
There is an associator, i.e. a natural isomorphismα with components, αA,B,C :pAbBq bCÑAb pBbCq
induced from the homotopyH:IIÑI
Hpx, tq $ & % xpt 1q ifxP r0,14s x 4t ifxP r14,12s x 2tp1xq ifxP r12,1s.
The inverse corresponds to replacingt by 1tin the homotopy.
The unit object 1 P ApIq is defined as p1 from the unique map I ÝÑp pt and
the unique element 1 P Apptq. 1 acts as left and right identity, i.e. there are two natural isomorphisms λ and ρ, with components λA : 1bA A and
ρA : Ab1 A. We abuse notation, denoting each unit element 1 to avoid overuse of subscripts. We defineρA:Ab1A as the morphism induced from
ρ:I2 ÑI:px, tq ÞÑmaxtp1 tqx,1u
which is a homotopy from the identity to maxt2x,1u : I Ñ I. We define λA
similarly.
Lemma 9.5. The mapsApIqÝÑ ApIqinduced byxÞÑ1xandApI2qÝÑ ApI2q
induced by px, yq ÞÑ p1x, yq define a dual for each object and morphism re- spectively.
Proof. To prove this lemma we must construct for each object A, a unit ηA : 1ÑAbA,a counitA:AbAÑ1, and we must verify the triangle identities,
i.e. verify that the composites
AÝÝÝÝÑηAbA AbAbAÝÝÝÑAbA A and AÝÝÝÝÑAbηA AbAbA ÝÝÝÝÑAbA A
are identities. Given A P ApIq we want an element of ApI2q such that the
restriction to the edges gives elements of ApIq corresponding to the labelling in
Figure 9.2. This will be the unit corresponding to A,i.e. ηA.
Figure 9.2: Schematic diagram of the boundary conditions on the object in Strat2 which corresponds to the unit element ηA, labelled with the elements of
ApIq obtained by restrictingA to those edges. Here and in what follows y0
is at the top of the diagram.
Take the mapF :I2 ÑIgiven by
Fpx, yq $ & % 12p b px 12q2 py1q2q if b px12q2 py1q2 ¤ 1 2 0 if b px12q2 py1q2q ¥ 1 2,
9.1 Dagger Pivotal Cat from a One-Object Whitney 2-Cat
Figure 9.3: We mark a half-disk. F maps each radial line segment on toA, with the arrows showing orientation. The boundary of the disk, the remaining edges of the square and the shaded region all get mapped to 0.
see Figure 9.3. This map is continuous and, by the Whitney Approximation Theorem [14,§10], it is homotopic to a smooth map ˜F :I2 ÑI. Since the map
F is smooth everywhere except the point p12,1q and the set
tpx, yq| c px1 2q 2 py1q2 1 2u,
we can choose a homotopy relative to the closed complement of an open neigh- bourhoods of these sets. We can choose a smoothing so that ˜F is prestratified and is symmetric about the linex 12. Moreover one can choose the homotopy so that it is through prestratified maps. After the smoothing process we have an induced map r F:ApIq ÑApI2q:AÞÑ p1Ñ AbAq. But, AbA A˜b p rAq AbA. (9.1) The first equality comes from the fact that we chose a symmetric smoothing and the isomorphism (of objects, in the usual categorical sense) from restricting the homotopy fromF to ˜F. We define
r
FA ηA: 1Ñ AbA
and compose with the isomorphism in (9.1) to getηA.
Next we verify the triangle identity for ηA ( can be constructed similarly). To
do this we need a homotopy between two mapsI2 ÑI.Rather than write down
an explicit homotopy we direct the reader to Figure 9.4. In it morphisms are composed via stacking. We can choose a homotopy that ‘straightens’ the picture