MAPPING SPACES OF Gray -CATEGORIES

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MAPPING SPACES OF Gray -CATEGORIES

BJÖRN GOHLA

Abstract. We dene a mapping space forGray-enriched categories adapted to higher gauge theory. Our construction diers signicantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Gray-categories and a kind of comonadic resolution of the 1-dimensional structure of a given Gray-category obtained by lifting the resolution of ordinary categories along the canonical bration ofGrayCatoverCat.

1. Introduction

It is well known that among algebraic models for homotopyn-typesGray-groupoids model 3-types; Lack [2011] gives us a proof using model category methods. Wanting to study the homotopy 3-type of the moduli space of 3-connections on a manifold, we thought it apt to dene a mapping space [S3(M),C(H)] of Gray-groupoids that could model that moduli space, where S3(M) is the fundamental Gray-groupoid and C(H) is the Gray- groupoid ultimately derived from a 2-crossed Lie-algebra where the triconnections take their values; see for example Schreiber and Waldorf [2011] for 2-connections, to which

The author was supported by FCT (Portugal) through the doctoral grant SFRH/BD/33368/2008.

This work was supported by FCT, with European Regional Development Fund (COMPETE) and na- tional funds, by means of the projects PTDC/MAT/098770/2008 Invariantes Topológicos via Geometria Diferencial and PTDC/MAT/101503/2008 Nova Geometria e Topologia. The author is a member of CMUP/Universidade do Porto. The hospitality of CMA/Universidade Nova de Lisboa is gratefully ac- knowledged.

Received by the editors 2013-02-17 and, in revised form, 2014-04-15.

Transmitted by Ieke Moerdijk. Published on 2014-04-17.

2010 Mathematics Subject Classication: primary: 18D05, 18D20; secondary: 55Q15 . Key words and phrases: Higher gauge theory, Gray-categories.

c

Björn Gohla, 2014. Permission to copy for private use granted.

100

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this is an obvious next step. See [Martins and Picken 2011] for the background on the smooth fundamentalGray-groupoid and triconnections. Wang [2013] shows how to obtain the gauge 3-connection from a 3-holonomy, and the 3-gauge transformation from the lax-transformation between holonomy Gray-functors.

The original denition of the Gray-tensor can be found in [Gray 1974]; Gordon et al.

[1995] give us the denition of tricategories and show that every tricategory stricties to a triequivalent Gray-category. Crans [1999] gives an explicit, elementwise denition of Gray-categories.

In 1999 Crans gave a partial solution the mapping space problem; however, the ab- sence of an interchange law inGray-categories prevents lax transformations betweenGray- functors from being composable in general. The slightly unsatisfactory solution is to restrict to those transformations and higher cells that can in fact be composed; this does give a mapping space Gray-category, but a mere stopgap not sucient for our purposes.

Instead, we enlarge the repertoire of maps, and thereby transformations, in a way that will permit forming all composites of transformations; specically we introduce a 2-cocycle that intermediates coherently between the two possible evaluations of arrangements of squares shown in (36) and (37). In analogy with Garner [2010] we introduce a co-monadic weakening of strict Gray-functors in section 2. The comonad Q¹ then yields a co-Kleisli category GrayCat_Q¹. We use in an essential way that GrayCat is bered over Cat.

Inspired by [Bénabou 1967] we axiomatise lax transformations as maps into a path- space. In section 3 we introduce a functorial path-space construction for Gray-categories;

subsequently in section 4 we show that this yields an internal category−→

H^−→_−→HinGrayCat_Q¹ for a given H in GrayCat.

The n-th iterate of −→

(_) yields an n-truncated internal cubical object in GrayCat. In section 5 we construct an internalGray-category

H^−→−→H^−→−→

−

→ H^−→−→H

in GrayCat_Q¹ as a subobject of the third iterated path-space. It is then a trivial consequence in section 6 that we obtain a mapping space Gray-category by applying the hom functor

[G,H] := GrayCat_Q¹(G,H^−→_−→H^−→_−→−→ H^−→_−→H).

Furthermore we obtain a restricted mapping space{G,H}, where everything is as before, except only strict Gray-functors are permitted between G and H. This leads to a natural sesquicategory structure on GrayCat.

We hope to be able to prove in a later paper that this internal hom is part of a monoidal closed structure on GrayCat_Q¹ involving a suitable extension of Crans' tensor product.

Finally, in section 7 we give explicit details of functors, transformations and so on in terms of components. Lastly, we remark that if H is a Gray-groupoid then −→

H as well as [G,H]will be Gray-groupoids.

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Similar work was done by Gohla and Martins [2013] concerning 2-crossed modules, which are equivalent to Gray-groupoids with a single vertex, that is, Gray-groups.

A version of this article constituted the author's doctoral thesis defended at the Faculty of Science, University of Porto. Many thanks are owed to João Faria Martins for plentiful advice and discussion.

2. Resolution in Dimension One

We dene a resolution of the 1-dimensional structure of aGray-category using a comonad, by lifting the free category comonad (called path in [Dawson et al. 2006]) to Gray- categories; but note that we use the term in a dierent way in this paper.

The resulting co-Kleisli category can be seen as the category ofGray-categories with an enlarged repertoire of maps, that is exible enough to carry out our path space construction. After giving an abstract construction of this category of pseudo maps we proceed to characterize them explicitly.

2.1. Basic Fibrations. There are obvious functors

GrayCat ⁽^_⁾²^//SesquiCat ⁽^_⁾¹^//Cat ⁽^_⁾⁰^//Set

that forget the 3-cells, the 2-cells and 1-cells respectively. By a slight abuse of language we will denote the composite (_)₁(_)₂ by (_)₁ also, it is of course a bration as well;

we will use it in section 2.12 to construct the monad Q¹. We will use the bration (_)₂(_)₁(_)₀ = (_)₀ in section 6 to construct the restricted mapping space {G,H}.

Let S be a sesquicategory,G a Gray-category, and F: S−→G2 a sesquifunctor. We dene F:F^∗S−→G as follows:

(F^∗S)₀ =S₀ (F^∗S)₁ =S1

(F^∗S)2 =S2

(F^∗S)₃ ={(Γ;α, β) Γ : F α−→F β}

Note that the interchange of two 2-cells α, β in F^∗S incident on a 0-cell is given essentially by the interchange of their images under F:

β⊗α= (F β⊗F β;β . α, β / α).

Let us take note of the following useful fact that helps to characterize the Cartesian maps:

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2.2. Remark. For a functor p: E −→ B that preserves co-limits, let D: D −→ E a diagram in E with co-limit (C, k_i)

D_i ^kⁱ ^//C

g

''A

f //B ,

assume p(g) factors below as p(f)u = p(g). Furthermore, assume that the induced sink (ui) =up(ki) has llers huii above with fhuii=gki, then the co-universally induced map hui: C −→A is a ller over u.

This means that to check whether a map f is Cartesian we don't need to give the ller u directly, but we can dene it on presumably simpler parts of C. These then combine into a valid ller.

2.3. Remark. Maps Cartesian with respect to (_)₂ are exactly the Gray-functors, that are 2-locally isomorphisms of sets. That is, given two parallel 2-cells on the intervening 3-cells, the map is bijective.

2.4. Lemma. F^∗S is a Gray-category, F is a Gray-functor and Cartesian with respect to

(_)2.

Similarly, let S be a sesquicategory, C a category, and F: C −→ S₁ a functor, then we dene a sesquicategory:

(F^∗C)₀ =C₀ (F^∗C)₁ =C₁

(F^∗C)2 ={(α;f, g) α:F f −→F g}

2.5. Lemma. F^∗C is a sesquicategory, F is a sesquifunctor, and Cartesian with respect

to (_)₁.

2.6. Remark. Maps Cartesian with respect to (_)₁ are exactly the sesquifunctors, that are 1-locally isomorphisms of sets. That is, given two parallel 1-cells on the intervening 2-cells, the map is bijective.

For later reference we describe the Cartesian liftings of (_)1 explicitly as well. Let G be a Gray-category, G1 its underlying category. Let C be an ordinary category and F: C−→G1 a functor. Then F^∗G is given by:

(F^∗G)₀ =C₀ (F^∗G)₁ =C₁

(F^∗G)₂ ={(α;f, g) f, g: x−→y, α: F f −→F g}

(F^∗G)₃ ={(Γ;α, β;f, g) f, g: x−→y, Γ : F α−→F β}

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Source and target maps are as follows:

s2(Γ;α, β;f, g) = (α;f, g) t2(Γ;α, β;f, g) = (β;f, g)

s₁(α;f, g) =f t₁(α;f, g) = g .

and s₀, t₀ are as given by C. As identities we take:

i₁(f) = (id_{F f};f, f) i₂(α;f, g) = (id_α;α, α, f, g). The tensor in F^∗Gof two 2-cells is

(β;g, g⁰)⊗(α;f, f⁰) = (β⊗α;β / α, β . α;g#₀f, g⁰#₀f⁰) (1) where

β / α= (β#₀F f⁰)#₁(F g#₀α), β . α= (F g⁰#₀α)#₁(β#₁F f).

There is an obvious map F: F^∗G−→GoverF that acts like F on 0- and 1-cells, and on 2- and 3-cells as a projection to G.

2.7. Remark. The globular set F^∗G is a Gray-category. The composition operations of F^∗G are given by those of C and G and it is easy to see that they fulll the axioms of a Gray-category.

Obviously G^∗F^∗G ∼= (F G)^∗G and id^∗_C ∼= id_GrayCat_C coherently. Also, we can always choose id^∗_C = id_GrayCat_C, but this is not necessary in what follows.

2.8. Lemma. A map of Gray-categories is Cartesian with respect to G 7→ G¹ i it is 1- locally an isomorphism of categories, i.e. given two parallel 1-cells the map is bijective on the intervening 2-cells and in turn bijective on the 3-cells between parallel such.

2.9. Definition. We dene a map of Gray-categories to be an n-isomorphism if it is Cartesian with respect to (_)_n. It is n-faithful if llers of factorizations under (_)_n are unique, and n-full is there (not necessarily unique) llers for all factorizations under (_)_n.

By this denition 0-delity is ordinary delity of functors, 1-delity is local delity, and so on.

2.10. Remark. One property of Cartesian maps in a bration p that we are going to exploit in the proof of the following theorem is that for three arrows upstairs,

r //

s // ^f //

with f Cartesian, p(r) = p(s) downstairs and f r =f s upstairs imply r =s, on account of f being p-faithful.

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2.11. Lemma. If f g is Cartesian with respect to a given bration p and f is p-faithful, then g is p-Cartesian.

Proof Take k and u such that p(g)u = p(k), then p(f g)u = p(f k) and hence by f g being p-full there is a ller hui such that f ghui = f k. Then by f being p-faithful ghui=k.

By f g being p-faithful hui is the unique such ller.

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2.12. Comonad Liftings. In this section we show that comonads can be lifted along brations of categories.

2.13. Definition. In an arbitrary 2-category a comonad on an object A is given by an endomorphism

A ^T ^//A and 2-cells

A

T

A

DDA

ε

and

A T //

T

A

T //A

δ

such that

A ^T ^//

T

A

T ##

A

==A

ε

δ = A

T

DDA

T =

A

T

##

A

==A ^T ^//A

ε

δ

and

A T //

T

A T //

T

A T //A

δ _δ

=

A T //

T

A T //A

T //A

δ

.

See, for example, Mac Lane [1998].

IfAis a category,T a functor andεandδnatural transformations, then these equations of course amount to the usual equations objectwise in A:

T x

{{

T x

##δx

T x T T x

T εx

//εT x

oo T x

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and

T x ^δ^x ^//

δx

T T x

T δx

T T x

δT x

//T T T x .

2.14. Theorem. Given a bration of categories p: E −→ B, a comonad (Q, δ, ε) on B can be lifted to a comonad (K, d, e) on E such that (K, Q) : p−→ p is a comonad in the 2-category of all brations.

Proof Let (_)^∗: B^op −→ Cat be a chosen cleavage. For every A ∈ E_x we let e_A: (KA =ε^∗_xA)−→ A be the chosen Cartesian lift of ε_x: Qx −→x. For a morphism f over j in

KA ^e^A ^//

Kf ##

A

f

!!KB _e

B //B

Qx ^ε^x ^//

Qj ""

x

j

!!Qy _ε

y //y

the dotted arrow is the unique ller induced by the factorization below. This makes K a functor and e: K −→idE a natural transformation.

We dene a family of co-multiplication maps d_A as the unique llers in KA

KA

))dA $$

KKA _e

KA//KA

Qx

))δx ##

QQx _ε

Qx

//Qx

where the triangle below commutes because Qis co-unital.

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In the diagram

KA

))dA $$

KA

eA

KKA ^Ke^A ^//

eKA

//KA _e

A //A

Qx

δx ##

Qx

εx

QQx ^Qε^x ^//

εQx //Qx _ε

x //x

we see that e_Ae_KAd_A = e_AKe_Ad_A by the naturality of e, and p(e_KAd_A) = p(Ke_Ad_A) by Q being a comonad. Hence by remark 2.10 the three endomorphisms of KA above have to coincide, meaning d is co-unital component wise.

The naturality of d, that is, that d_BKf = KKf d_A is the unique ller making the left-hand upstairs square commute

KA

**

dA //

Kf $$

KKA

KKf

%%KB

dB

//KKB _e

KB

//KB

Qx ^δ^x ^//

Qj ##

QQx

QQj

$$

Qy δy

//QQy _ε

Qy //Qy

is obtained by observing thate_KBd_BKf =KF =Kf e_KAd_A=e_KBKKf d_A, fromebeing natural and a retraction. Also,p(d_BKf) =p(KKf d_a)by naturality of δ. We apply 2.10 again.

Finally, we show that d is co-associative: Consider the diagram KA

**

dA //

d_A $$

KKA

dKA

&&

KKA KdA

//KKKA_e

KKA

//KKA

Qx ^δ^x ^//

δx ##

QQx

δQx

%%

QQx Qδx

//QQQx _ε

QQx//QQx .

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We calculate that e_KKAKd_Ad_A = d_Ae_KAd_A = d_A = e_KKAd_KAd_A, again by naturality of e and its retractiveness. Moreover, δ is co-associative, hence we can apply remark 2.10

once more.

We observe thatK preserves Cartesianness of maps, thus in particularKeis Cartesian component wise.

Finally we can dene our resolution comonad. Let (Q, δ, ε) = (F U, F ηU, ε) be the comonad that arises from the adjunction

RGrph

F ++

Cat

U

mm _ .

Then, according to theorem 2.14, we obtain the comonad (Q¹, d, e) on GrayCat induced by lifting Q along (_)₁. The exponent reminds us that this provides a resolution of the 1-dimensional structure of Gray-categories. See section A for a more abstract point of view on this construction. In section 2.22 we will show explicitly how this comonad acts.

2.15. Corollary. By the above theorem there is a comonad Q¹ on GrayCat that pulls back the Gray-structure onto the free category on the underlying 1-graph.

If a categoryCis already the free categoryC =Fgover a reexive graph with injection of generators η: g−→UC, then by adjointness the counit is split

C ^{F η} ^//

C

??QC ^ε ^//C

.

2.16. Definition. If a Gray-category G has an underlying category G1 of the form Fg for some reexive graph g we say that G is free up to order 1 with generating 1-cellsg. Let k: G−→Q¹G be the ller along (_)₁ for the factorization e₁F η = (id_G)₁ for the given generating reexive graph. This of course gives a splitting

G ^k ^//

G

==Q¹G ^e ^//G

. (2)

If a Gray-category is free up to order 1 we may look at the 1-cells as follows: every 1-cell f can be written as [f₁, . . . , f_n], where the [f_i] are generating 1-cells unique up to insertion and deletion of units. Now, the action of k: G −→ Q¹G can be described as follows:

1. 0-cells: k: x7→x

2. 1-cells: k: f = [f₁, . . . , f_n]7→[[f₁], . . . ,[f_n]]

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3. 2-cells: k: (α: f =⇒f⁰)7→(α; [[f₁], . . . ,[f_n]],[[f₁⁰], . . . ,[f_n⁰0]]) 4. 3-cells: k: (Γ : αVα⁰)7→(Γ;α, α⁰; [[f₁], . . . ,[f_n]],[[f₁⁰], . . . ,[f_n⁰0]]) This is obviously a section of e_G.

2.17. Definition. The category of Gray-categories and pseudo Gray-maps is the co- Kleisli-category GrayCat_Q¹ of the comonad Q¹.

2.18. Lemma. The map k for a G free up to order 1 has the following nice behaviour with respect to Q¹:

G ^k ^//

k

Q¹G

d

Q¹G _Q1k

//Q¹Q¹G

. (3)

commutes.

Proof We apply remark 2.10: The diagram G ^k ^//

k

Q¹G

d

Q¹G _Q1k

//

e

Q¹Q¹G

e

G _k ^//Q¹G

commutes by co-unitality and the denition ofk. Also under(_)₁the diagram (3) becomes Fg ^{F η} ^//

F η

F U Fg

F ηU F

F U Fg

F U F U η//F U F U Fg

which commutes by naturality of η.

This category has Gray-categories as objects, and morphisms G

f //H are morphisms Q¹G

f //H

inGrayCat. Composition of two maps G

f //H

g //K

is dened by

Q¹G ^d^G ^//Q¹Q¹G

Q¹f//Q¹H

g //K.

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Identities are of the form

G ^id^G ^//G = Q¹G ^e^G ^//G.

By way of notational convenience in diagrams in GrayCat_Q¹ we use unslashed arrows f:G−→H to denote a strict arrow that is included in GrayCat_Q¹ asf e: G 9 H.

The comonad axioms make sure this is a category; c. f. e. g. [Mac Lane 1998].

There is an adjunction

GrayCat

R ..

GrayCat_Q¹

L

mm _

The functor R takes a strict map f: G −→ H to a pseudo map f e: G 9 H where e is the co-unit of Q¹. Moreover, since e is an epimorphism, R is faithful, and it is bijective on objects, hence R is actually an inclusion; in particular, we have injective maps

GrayCat(G,H) ^e^∗ ^//GrayCat_Q¹(G,H) (4) for all G and H.

We note that the composite of a strict map after a pseudo map is particularly simple:

G ^f ^//H ^ge ^//K = Q¹G

d_Q1

G//Q¹Q¹G ^Q

1f //

e_Q1

G

Q¹H ^ge ^//

e_H

K

Q¹G _f ^//H

g

== . (5)

If G is free up to order 1 we also get an idempotent function GrayCat_Q¹(G,H) ^(ke)

∗//GrayCat_Q¹(G,H) (6) from (2) we might call strictication (note the reverse order of k and e). It preserves the image of the functor R, that is, strict Gray-functors are preserved.

2.19. Lemma. The category GrayCat_Q¹ has all limits of diagrams of strict maps, that is, those in the subcategoryGrayCat, that is,GrayCatis complete and the inclusionGrayCat−→

GrayCat_Q¹ preserves all limits.

Proof Let D be a diagram in GrayCat, let (`_i: L −→ D_i)_i be a limiting source in GrayCat, we claim its embedding into GrayCat_Q¹ is a limiting source there as well.

Let(ci: C 9Di)ibe a source overDinGrayCat_Q¹. Thus there is a source(ci: Q¹C −→

D_i)_i in GrayCat, which induces a map hci: Q¹C −→ L and this is of course a map hci: C 9L. The diagram

C

ci

r

hciU

L

`i

//D_i

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commutes for all i by the co-unit axiom of Q¹ and the naturality of e; c. f. also (5).

Because e is an epimorphism hci is the unique ller.

In particular, the pullback of two strict maps inGrayCat_Q¹ is the same as its pullback in GrayCat. Products are obviously simply the same in both categories since their diagrams do not include any nontrivial morphisms.

2.20. Remark. For two diagrams {a_k: Gi −→ Gj}, {b_k: Hi −→ Hj} of strict maps of the same type in GrayCat_Q¹ and a natural transformation f_i: Gⁱ 9 Hⁱ between them there is an induced map lim{f˙ _i} such that:

lim{Gi, a_k} ^limf^˙ ⁱ//

pi

lim{Hi, b_k}

p⁰_i

Gi

fi

//Hi

. (7)

We unravel this diagram in terms of maps in GrayCat and obtain

Q¹lim{Gⁱ, a_k}

limf˙ i

))

Q¹pi ))

h^Q¹^pii_//

lim{Q¹Gⁱ,Q¹a_k}

ri

limfi //lim{Hⁱ, b_k}

p⁰_i

Q¹Gⁱ _f

i

//Hⁱ

where the maplimf˙ _i is induced by the universal property of the source{f_iQ¹p_i}inGrayCat, that is,lim{f˙ i}=hfiQ¹pii, which then is the appropriate map inGrayCat_Q¹. On the other hand, limf_i is induced by the cone f_ir_i. By universality limf˙ _i = limf_ihQ¹p_ii.

In particular this applies to pullbacks, that is, there is a canonical map f×g˙ :G×_KH 9 G⁰×_K⁰H⁰

determined by f, g, h in

H

a

g //H⁰

a⁰

G

b

f //G⁰

b⁰

K _h ^//K⁰

. (8)

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2.21. Remark. If in (7) the maps f_i are of the form g_ie, i.e. the f_i come from strict maps, then we have

lim(g˙ _ie) = (limg_i)e . In particular in a situation analogous to (8) we have

(f e) ˙×(ge) = (f×g)e (9)

2.22. Special Cells in the Resolved Space. We now take a closer look at the structure ofQ¹G. By denition 1-cells here are non-empty lists [f1, . . . , fn]of composable G-1-cells modulo insertion or removal of identity 1-cells of G; composition is concate- nation. For composable 1-cells in G, say, f₁, . . . , f_n we have several 1-cells in Q¹G, in particular [f1, . . . , fn] = [f1]#0· · ·#0[fn] and [f1#0· · ·#0fn] and e_G maps all of these to f₁#₀· · ·#₀f_n. Between [f₁, . . . , f_n] and [f₁#₀· · ·#₀f_n] we have a 2-cell

κf1,...,fn = (idf1#0···#0fn; [f1, . . . , fn],[f1#0· · ·#0fn])

that is the pulled back identity 2-cell of f₁#₀· · ·#₀f_n. In particular we have

[f2] //

[f1#0f2]

[f1]

κf1,f2

{

for all for all pairs f₁, f₂ of 1-cells of G. Whiskers and composites of higher cells in Q¹G are simply carried out in G, hence for example

κ_f₁_,f₂#₀[f₃] = (id_f₁_#₀_f₂#₀f₃; [f₁, f₂]#₀[f₃],[f₁#₀f₂]#₀[f₃])

= (id_f₁_#₀_f₂_#₀_f₃; [f₁, f₂, f₃],[f₁#₀f₂, f₃]) and

κ_f₁_#₀_f₂_,f₃#₁(κ_f₁_,f₂#₀[f₃]) = (id_f₁_#₀_f₂_#₀_f₃; [f₁, f₂, f₃],[f₁#₀f₂#₀f₃]) =κ_f₁_,f₂_,f₃. Hence we obtain that

[f₁]#₀[f₂]#₀[f₃] ^[f¹^]#⁰^κ^f²^,f³ ⁺³

κf1,f2#0[f3]

κf1,f2,f3

!)

[f₁]#₀[f₂#₀f₃]

κf1,f2#0f3

[f₁#₀f₂]#₀[f₃] _κ

f1#0f2,f3

+3[f₁#₀f₂#₂f₃]

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commutes.

We consider the possible horizontal composites of κ_f₁_,f₂ and κ_f₃_,f₄ and their tensor:

[f3,f4]

##

[f3#0f4]

;;

κf3,f4

[f1,f2]

##

[f1#0f2]

;;

κf1,f2

[f3,f4]

##

[f3#0f4]

;;

κf3,f4

[f1,f2]

##

[f1#0f2]

;;

κf1,f2

κf1,f2⊗κ_f

3*4,f4 .

By (1) we obtain

κ_f₁_,f₂⊗κ_f₃_,f₄ = (id_f₁_#₀_f₂; [f₁, f₂],[f₁#₀f₂])⊗(id_f₃_#₀_f₄; [f₃, f₄],[f₃#₀f₄])

=







id_f₁_#₀_f₂⊗id_f₃_#₀_f₄;

(id_f₁_#₀_f₂#₀e[f₃#₀f₄])#₁(e[f₁, f₂]#₀id_f₃_#₀_f₄), (e[f₁#₀f₂]#₀id_f₃_#₀_f₄)#₁(id_f₁_#₀_f₂#₀e[f₃, f₄]);

[f₁, f₂, f₃, f₄],[f₁#₀f₂, f₃#₀f₄]







=







id_id_f

1#0f2#0f3#0f4;

(id_f₁_#₀_f₂#₀f₃#₀f₄)#₁(f₁#₀f₂#₀id_f₃_#₀_f₄), (f₁#₀f₂#₀id_f₃_#₀_f₄)#₁(id_f₁_#₀_f₂#₀f₃#₀f₄);

[f₁, f₂, f₃, f₄],[f₁#₀f₂, f₃#₀f₄]







=







id_id_f

1#0f2#0f3#0f4;

(id_f₁_#₀_f₂_#₀_f₃_#₀_f₄)#₁(id_f₁_#₀_f₂_#₀_f₃_#₀_f₄), (id_f₁_#₀_f₂_#₀_f₃_#₀_f₄)#₁(id_f₁_#₀_f₂_#₀_f₃_#₀_f₄);

[f₁, f₂, f₃, f₄],[f₁#₀f₂, f₃#₀f₄]







=







id_id_f

1#0f2#0f3#0f4; id_f₁_#₀_f₂_#₀_f₃_#₀_f₄, id_f₁_#₀_f₂_#₀_f₃_#₀_f₄;

[f₁, f₂, f₃, f₄],[f₁#₀f₂, f₃#₀f₄]





 ,

meaning that this tensor is the identity of the two possible horizontal composites ofκ_f₁_,f₂ and κf3,f4.

Finally, note that by construction the κ_f₁_,...,f_n are all invertible.

2.23. Pseudo Maps Explicitly. We provide an elementary characterization of pseudo Gray-functors.

2.24. Definition. A pseudo Q¹ graph map F: G −→ H between Gray-categories is a map of 3-globular sets, together with a function F²: G¹ ×_G₀ G¹ −→ H², such that the following conditions hold:

1. the restriction of F to G(x, y) is a sesquifunctor for all 0-cells x, y of G,

2. F²is a normalized 2-cocycle, that is, theF_f²₁_,f₂ are invertible 2-cellsF_f²₁_,f₂: F(f₁)#₀F(f₂) =⇒ F(f₁#₀f₂) with

F_f²₁_,f₂_#₀_f₃#1(F(f1)#0F_f²₂_,f₃) = F_f²₁_#₀_f₂_,f₃#1(F_f²₁_,f₂#0F(f3)), (11)

(16)

and for f₁ or f₂ an identity 1-cell we have

F_f²₁_,f₂ = id_{F f}₁_#₀_{F f}₂,

3. left and right whiskers of 2-cells by 1-cells along 0-cells are coherently preserved:

F(α#₀f)#₁F_g,f² =F_g²0,f#₁(F α#₀F f) (12) F(g#₀β)#₁F_g,f² =F_g,f² 0#₁(F g#₀F β)

4. left and right whiskers of 3-cells by 1-cells along 0-cells are coherently preserved:

F(Γ#₀f)#₁F_g,f² =F_g²0,f#₁(FΓ#₀F f) (13) F(g#₀∆)#₁F_g,f² =F_g,f² 0#₁(F g#₀F∆)

5. the tensor is coherently preserved:

F(β⊗α)#1F_g,f² =F_g²⁰_,f⁰#1(F β⊗F α) (14) 6. the tensors of compositors are trivial:

F_f²₁_,f₂ / F_f²₃_,f₄

F_f²

1,f2⊗F_f²

3,f4*4F_f²₁_,f₂ . F_{f e}²₃_,f₄

!

= id (15)

7. tensors of 2-co-cycle elements with images of 2-cells vanish:

F α / F_g,f²

F α⊗F_g,f²

*4F α . F²_g,f

!

= id (16)

F_h,g² / F β ^F

2

h,g⊗F β

*4F²_h,g. F β

!

= id (17)

for all suitably incident cells. Denote the set of all pseudo Q¹-graph maps from G to H by M(G,H).

Note also how the identity 1-cells of a 0-cells are preserved strictly, this is part of the globularity condition.

Note furthermore how this denition implies that the horizontal composites are also coherently preserved as a consequence of (12):

F(α / β)#₁F_g,f² =F_g²0,f⁰#₁(F α / F β) F(α . β)#₁F_g,f² =F_g²0,f⁰#₁(F α . F β).

(17)

2.25. Lemma. There is a canonical correspondence between the set of pseudo Q¹ graph maps M(G,H) and co-Kleisli maps GrayCat_Q¹(G,H).

M(G,H)

(_)^˜

&&

GrayCat_Q¹(G,H)

(_)^∨

ff

Proof Given a Q¹ graph map F: G−→ H we dene a Gray-functor F˜: Q¹G−→H as follows

1. 0-cells:

F˜(x) = F(x), 2. 1-cells:

F˜[f₁, . . . , f_n] =F f₁#₀· · ·#₀F f_n, 3. 2-cells:

F˜(α; [f₁, . . . , f_n],[g₁, . . . , g_m]) = ˜F κ_g₁_,...,g_m#₁F α#₁F κ˜ _f₁_,...,f_n (18) where for n= 2the 2-cell F κ˜ _f₁_,...,f_n is dened as F_f²

1,f2 and for n ≥3as the unique extension due to (11), (15),

4. 3-cells:

F˜(Γ;α, β; [f₁, . . . , f_n],[g₁, . . . , g_m]) = ˜F κ_g₁_,...,g_m#₁FΓ#₁F κ˜ _f₁_,...,f_n. To elucidate, we show that 1-2-whiskers are preserved by F˜. For whiskerable cells

[f1,...,fn] //

[g1,...,gm]

##

[g⁰₁,...,g⁰_m0]

;;

(β;...)

(18)

the equation

F˜[f1,...,fn]//

F˜[g1,...,gm]

##

F˜[g₁⁰,...,g⁰

m0]

;;

F˜(β;...)

= ^{F f}¹^#⁰^···#⁰^{F f}ⁿ ^//

F g1#0···#0F gm

F(g1#0···#0gm)

''

F(g⁰₁#0···#0g⁰

m0)

77

F g₁⁰#0···#0F g⁰

m0

GG

F κg1,...,gm

F β

F κ_g0 1,...,g0

m0

=

F g1#0···#₀F gm#0F f1#0···#₀F fn

F(g1#0···#₀gm#0f1#0···#₀fn)

++

F(g⁰₁#0···#₀g⁰

m0#0f1#0···#₀fn)

33

F g⁰₁#0···#₀F g⁰

m0#0F f1#0···#₀F fn

@@

F κg1,...,gm,f1,...,fn

F(β#0f1#0···#0fn)

F κ_g0 1,...,g0

m0,f1,...,fn

=

F˜([g1,...,gm]#0[f1,...,fn])

%%

F˜([g⁰₁,...,g⁰_m0]#0[f1,...,fn]).

99

F˜((β;...)#0[f1,...,fn])

is a consequence of (18).

Similarly, we can verify that F˜ preserves tensors: We calculate F˜((β; [g₁, . . . , g_m],[g₁⁰, . . . , g_m⁰ 0])⊗(α; [f₁, . . . , f_n],[f₁⁰, . . . , f_n⁰0]))

= ˜F (β⊗α;β / α, β . α; [g₁, . . . , g_m, f₁, . . . , f_n],[g₁⁰, . . . , g_m⁰ 0, f₁⁰, . . . , f_n⁰0])

= ˜F κ_g⁰

1,...,g⁰

m0,f₁⁰,...,f⁰

n0#₁F(β⊗α)#₁F˜_g₁_,...,g_m_,f₁_,...,f_n

= ( ˜F κ_g⁰

1,...,g⁰

m0⊗F κ˜ _f⁰

1,...,f⁰

n0)#₁(F β⊗F α)#₁( ˜F_g₁_,...,g_m⊗F˜_f₁_,...,f_n)

= ( ˜F κ_g⁰

1,...,g⁰

m0#₁F β#₁F˜_g₁_,...,g_m)⊗( ˜F κ_f⁰

1,...,f⁰

n0#₁F α#₁F˜_f₁_,...,f_n)

F˜(β; [g₁, . . . , g_m],[g⁰₁, . . . , g_m⁰ 0])⊗F˜(α; [f₁, . . . , f_n],[f₁⁰, . . . , f_n⁰0]) using (14) and (15). Preservation of the remaining operations is equally simple to verify.

Conversely, given a Gray-functor G: Q¹G −→ H we dene a pseudo Q¹ graph map Gˇ: G−→H as follows:

1. 0-cells: G(x) =ˇ G(x) 2. 1-cells: G(fˇ ) =G[f]

3. 2-cells: G(α) =ˇ G(α; [f],[f⁰]) 4. 3-cells: G(Γ) =ˇ G(Γ;α, β; [f],[f⁰])

MAPPING SPACES OF Gray -CATEGORIES