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Figure from:Classifying spaces for braided monoidal categories and lax diagrams of bicategories

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which take a (p, q)-simplex (65) of AZ? (M,@®,c) to the (p,q + 1)-simplex bicat which, for each p > 1, is given by the maps hm, 0 <_m < q, as in the diagram — Figure 55 which take a (p, q)-simplex (65) of AZ? (M,@®,c) to the (p,q + 1)-simplex bicat which, for each p > 1, is given by the maps hm, 0 <_m < q, as in the diagram

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will represent a deformation y whose source (resp. target) is obtained by horizontal composition of the morphisms in the string uo,...,Un (resp. Up,---,;Um) following a particular given association. By the bicategorical coherence theorem, such a deformation uniquely determines another when any other particular bracketing is used for computing the source and the target morphism from the given strings of morphisms. Therefore, diagram (1) is not ambiguous once a choice of association has been made for the source and target of the deformation. When F’: A > B is a homomorphism and diagram (1) is given in A, then we will denote by deformation it is. The lax functor is termed a pseudo-functor or homomorphism whenever all the structure constraints Ff are invertible. If the unit constraints F), are all identities, then the lax functor is qualified as (strictly) unitary or normal and if, moreover, the constraints Fy are also identities, then F' is called a 2-functor.

The horizontal composition of a modification yj: a= 8: F => F’ with a modi- fication wy’ : a! S 6: F’ => F" is the modification yw’ ow: a’ oa = Bo B such that (W’ ow)a = w'x o wa, for each object « of A. The structure constraints in Bicat(A, B) are canonically derived, in a pointwise way, from those of B; thus, for example, the associativity modifications a : a” 0 (a’ oa) > (a” 0a’) oa are given by AX = Aa''z,a'x,ax, LE ObA. Spine 4 ‘aan “ge fy aie ae my sg ne ae The composition of lax functors F': A 6 and G: B >C will be denoted by juxtaposition, that is, GF’: A — C. And recall that its constraints are obtained from those of F’ and G by the rules GF, = GFy»:Gru,ry and GF; = GF, -Gry. This composition of lax functors is associative and unitary, so that the category of bicategories and lax functors is defined. The composition of homomorphisms gives the function on objects of a homo- morphism of bicategories

respectively associated to pairs of composible arrows k 4 j “yi and objects i of J, and invertible modifications respectively associated to triplets of composible arrows £ > k a j 4 i and arrows j 4 i of the category I, subject to the two coherence axioms (CC1) and (CC2), as stated in Section 7.

and these are subject to the two coherence axioms (CC3) and (CC4), as stated in Section 7. for k j “+ i two composible arrows and j any object of I, the respective invertible modifications

is given by writing (m’ om); = mom, for each i € Ob/, while its component at an arrow a: j — 7 is the modification obtained by pasting the diagram The horizontal composition of J-modifications 0: m3 n:F => F' ando’:m 3S n’: F' => F", where F, F’, F” : F > G, is the I-modification o’00:m/om = n'on such that, for any object 7 of I, (a! 00); = a,oo;: mom; S nion,. All the structure constraints in the bicategory Bicat’'(F,G) are provided by using the corresponding structure constraints of the tricategory Bicat in a pointwise fashion. Thus, for example, For pseudo J-transformations F F’ 4 Pp" me FF": F 3G, the I-modification a : m” o (m’ om) =} (mom) om is that defined by the family of associativity modifications a : mi! o (miom;) > (mi! om‘) om, of Bicat(F;, 9), i € Obl.

and, finally, its component I at an object 7 of J is the modification obtained from those of F' and G, by pasting the diagram

And the composition of J-modifications ¢ : m =3 m!’: F => F’: F > G and Tinenr:GSGQ:GHisor:nmSn'm’, with (07); = oi s nim, S nimi, for every « € Obl. Moreover, given lax J-homomorphisms F 4, G SH and pseudo J-transformations F 3 F’ 4 F”:F >GandGS3Q@53G":GH, the structure constraint of the composition homomorphism (9), l¢r } lglr and n'm'onm = (n'on)(m' om), are provided by the family of modifications (4) le,r, > le,lr, and nimi onym: > (ni on;)(mi om), i € Obl, respectively. Thea aaanriatiwvity and wnit neandon natural aniivalencea

are as follows: For any lax J-homomorphisms F 5 g SH 4 K, H(GF) & (HG)F is the pseudo J-equivalence whose component at any object 7 of I is the identity on the homomorphism H;G;F;, and whose component at an arrow a: j — 7 is the modification obtained by pasting

The coherence triangles for unit constraints in lef follows from (CC2). The Grothendieck construction is natural on lax diagrams of bicategories. More precisely, we have the following:

So defined, [ 7 is indeed a pseudo transformation thanks to the coherence condi- tions (CC5) and (CC6). Us isvD beat S Olli Dil f If m,m': F + G: F — G are two pseudo J-transformations, it follows from the commutativity of the squares in (CC7) that every J-modification ¢ :m =m’ defines a modification which acts as follows on the hom-categories: Any pseudo J-transformation, m : F>G:F—3,G, gives rise to a pseudo-transformation

Ifm:F =>G: Ff 5G is any pseudo J-transformation, then Jplm = lpm and, for any lax J-homomorphism F': F —> G, the invertible structure constraint whose component at any pair of lax J-homomorphisms, F + G 4H, is the pseudo natural equivalence

are, respectively, the unique coherence 2-cells, (x, i) € Obf,F, For any given lax homomorphisms F + G Su T, the estructure invertible modifications w, 6 and y for [ ;, aS in the definition of a trihomomorphism,

Following Giraud [14], Street [33], Thomanson [35], and May [28], we shall show here how any lax J-diagram of bicategories F = (F, x, +,w, 7,6) : I°? > Bicat has, naturally associated to it, a genuine I[-diagram of bicategories, that is, a functor

and, for any object x in F;, the structure isomorphism 1j,, © Ji(1z) is that given by the canonical isomorphisms ux = 4x0 1g. For the horizontal composition of 1-cells z > y + x in F;, the structure invertible 2-cell Ji(u) o Ji(v) = Ji(wov) is provided by pasting the diagram in F;

For k j “4 i, two composible morphisms of J, and any object i, the invertible modifications

are respectively provided, at each object x of F;, by pasting the diagrams

Given F' : F > G, a lax I-homomorphism, for each object i of I, the pseudo- natural equivalence (19) at i,m; : F; J; > JiF; : F; > G;, is the identity on objects, that is, mix = 1(r,21,), while its component at a 1-cell y — x of the bicategory F;

is canonically obtained from pasting Finally, for a: 7 7a morphism of J, the corresponding invertible modification

and, for each object (x, i + 7), the structure identity constraint TL razr) 2 aly) is dg: YOa"t => lax.

whose structure constraints for horizontal compositions of 1-cells are given by the left identity constraints of the bicategories F. ' |,J can easier be described in terms of the following normal homomorphism Lemma 4.4. There is a pseudo-natural equivalence [,J > j: [, F > J, F

For the horizontal composition of 1-cells z “ y “yx in F;, the invertible structure 2-cell i(u) o i(v) = i(u ov) is provided by pasting the diagram (18) in F;, and, for any object x in F;, the structure isomorphism 1;, © i(1,) is the one given by the canonical isomorphisms 1,4 = 1,7 0 1,.

Proof. The square (33) is the composite of the squares where, in (a), both horizontal homomorphisms J; (17) and j (21) induce homotopy equivalences on classifying spaces, by Lemma 4.3, Proposition 4.5, and Fact 5.2 (iii). Therefore, the induced square (34) is homotopy cartesian if and only if the one induced by (b) is as well. But, recall that the rectification F : I’” > Bicat is a diagram, that is, a functor, and we have the natural homotopy equivalences (32). Therefore, it will be enough to prove that the induced pullback square of spaces

is homotopy cartesian, which is a consequence of Quillen’s Lemma [29, p. 90] (see also [15, §IV, Lemma 5.7]). To verify the hypothesis, simply note that, for each

moreover, €), =I and, for every arrow T:k — 1, the squares below commute. such that, for any three composible arrows in I, | Ak + j Si, the diagram below (where we have omitted the associativity constraints) is commutative

If €,€': X > N are 1-cells, then a 2-cell f : € = &' consists of a family of morphisms fo : &¢ + € in M, one for each arrow a: 7 +i in I, such that fy = 11 and for any two arrows k > j 2+ i the following square commutes:

which take a (p,q)-simplex (45) of A’'Ner(M, ®,c) to the (p,q + 1)-simplex which, for each p > 1, is given by the maps h,,, 0 < m < q, as in the diagram

while, for u = p and 7 < m < k, the corresponding (RimE Yi 5.4.1 is defined as the composite morphism

(CC7): for any arrow a: j > 7 of the category I, the square below commutes.

x of A, the component of a’ oa at a morphism u: « > y being the deformation obtained by pasting the diagram The horizontal composition of a modification yj : a= 8: F = F’ with a modi- fication wy’ : a! S 6’: F’ => F" is the modification y’ ow: a’ oa = Bo B such that (W’ ow)a = w'x o wax, for each object « of A. The structure constraints in Bicat(A, B) are canonically derived, in a pointwise way, from those of B; thus, for example, the associativity modifications a : a” 0 (a’ oa) > (a” oa’) oa are given by ax = Qo"'z,0'x,an, © E ObA. Sn ee ae: i en. | eee: ono Le A a, << ee eo Ww ee: See. oe ee: ee eee ie ee

and these are subject to the two coherence axioms (CC3) and (CC4), as stated in Section 7. for k j “4 i two composable arrows and j any object of J, the respective invertible modifications

Two I-modifications 0: m>n:F => F" and o’:m'3 7’: F’ = F” compose horizontally giving the [-modification o’0¢0:m'om=3n’on such that, for any object i of I, (c' 00); =a,00;: mom; > njion,;. All the structure constraints in the bicategory Bicat! '(F ,G) are provided by using the corresponding structure constraints of the tricategory Bicat in a pointwise fashion. Thus, for example, For pseudo J-transformations F 3 F’ 3 FF” 3 Fl” : F > G, the I-modification a:m"”o(m'om) => (m" om’) om is that defined by the family of associativity modifications a : mio (miom;) > (mi! om!) om, of Bicat(F;,G;), 1 € Obl. For lax [-diagrams of bicategories F, G, and H, the composition homomorphism one for each arrow a: j > i of I, subject to the two coherence conditions (CC5) and (CC6), as stated in Section 7. And, finally, say that if m,m’: F > F’: F > F' are pseudo J-transformations, then an I-modification 0 :m = m’ is a family of modifications

and, finally, its component I at an object 7 of J is the modification obtained from those of F' and G, by pasting the diagram

Ifm:F => F’:F —>Gandn:G>G':G —>H are pseudo J-transformations, then their composition is nm : GF = G’F’ : F + H, whose component at an object 7 is nym; : GF; => GF! : F; > H;, and whose component at an arrow a:j —7is the modification obtained by pasting the diagram

Besides, for any pseudo J-transformations (m,n,t) : (H,G,F) = (H’,G’, F’), the corresponding I-modification @ : @,, 61, 0 m(nt) > (mn)toa,¢ p, is given by the family of modifications (9) @ : 1 o m;(nit;) S (mini)t; 01, 7 € Obl. For any I-homomorphism F: F > G,l:1lgF => F andr: Fl¢ = F, are the pseudo I-equivalences whose components at any object 7 € Ob/ are both the identity on the homomorphism Fj and, at an arrow a: j — i, are the canonical isomorphism a*1p, 060g = 0a 0 lpia. Besides, form: F + F" any pseudo I-transformation, the corresponding T-modifications 1 : [rroligm > molp and Tirpoml, > morr are respectively given by the family of modifications (8), |: 1 F/Olig. m > molp, and? : lp omilis, >mo 1p. are as follows: For any lax J-homomorphisms F +> g SH 4 kK, H(GF) S$ (HG)F is the pseudo J-equivalence whose component at any object 7 of I is the identity on the homomorphism H;G;F;, and whose component at an arrow a: j — 7 is the modification obtained by pasting

The identity morphism, for each object (a,7) in {, F, is provided by the pseudo- transformation u:1z¢, > 17, by The structure associativity isomorphism The left and right identity constraints

The coherence pentagon for associativity inf. ;7 holds thanks to the coherence condition (CC1) in Section 7, and the coherence triangles for unit constraints in J, F follows from (CC2). Hence [, F is actually a bicategory.

So defined, [ ,;m is indeed a pseudo transformation thanks to the coherence condi- tions (CC5) and (CC6). If mm’ :F>G:F 5G are two pseudo J-transformations, it follows from is given by pasting tions (CC5) and (CC6). If m,m': F + G: F — G are two pseudo J-transformations, it follows from the commutativity of the squares in (CC7) that every J-modification ¢ :m =m’ defines a modification

Ifm:F > G:F —-G is any pseudo J-transformation, then frelon = lpm and, for any lax J-homomorphism F': F > G, the invertible structure constraint

3.3. The bicategorical Grothendieck construction as a tricolimit. In [17, § 8], Gray proved that the functor f; : Cat!” + Cat carries any lax J-diagram of categories to its lax colimit (or 2-colimit) in the 2-category of small categories. Next we shall prove the parallel fact for lax diagrams of bicategories. To do that, fix J any small category and let

and, for any object x in F;, the structure isomorphism 1,,, = ;(1,) is the one given by the canonical isomorphisms 44.7 = 4;201,. Ifa: 7 +7 is any morphism in I, then the component at an object « € F; of the attached pseudo transformation where, for the horizontal composition of 1-cells z Yy “, x in F;, the invertibl structure 2-cell 7;(u) om (v) & ni (u ov) is provided by pasting in F; the diagram

and, for a: 7 > i, the corresponding invertible modification is, at any object x of F;, that canonically obtained from pasting in G;

The tritransformation 7 takes any pseudo J-transformation m : F > F’: F +G to the invertible [-modification

since we have the invertible trimodification t and the J-trimodification s canonically obtained from the (triangulators) invertible trimodification T and I- trimodification S

described as follows: The component of T’ at any lax J-diagram J, is the pseudo equivalence T: € [,F [,nF > 1p with T(x,i) = l,i) for any object (x, 7) of J,;F, and whose component at a morphism (u,a) : (a,7) — (y,j) is canonically provided by the 2-cell in F; pasted of

and, for each object (x,i + 7), the identity structure constraint litea) = Ri(liz,a) is da: XO a => lax. The unit of the biadjunction is the pseudo-transformation 7 : lz, > RiJi, with ne = 4x : x — 1a, for each object « of F;, and whose component at a 1-cell u:y— ax is the invertible 2-cell obtained by pasting

The invertible modification triangulators lp, > Rjeo nR,; and €J; 0 Jin => 1, are, at objects («,i + 7) of F, and x of F;, respectively obtained from pasting the diagrams below in F;. provided from pasting in the bicategory F;

For any lax diagram of bicategories F : [°P — Bicat, the lax J-homomorphism J : F -»F induces a corresponding homomorphism on the Grothendieck construc- tions fpJ: f,F > J, F . Uptoapseudo-natural equivalence, this homomorphism {_J can easier be described in terms of the following normal homomorvhism

whose structure constraints for horizontal compositions of 1-cells are given by the left identity constraints of the bicategories F, . J,J can easier be described in terms of the following normal homomorphism

such that, for any three composible arrows in I, | 3 k 5 j 4 i, the diagram below (where we have omitted the associativity constraints) is commutative whose 1-cells €: \ > A’ consist of pairs of maps assigning

For any n > 2, we have the equality P,,E, = 1 and, moreover, there is a oplax transformation, whose component at a 3-cocycle A : [n] > (M,®,c) is the 3-cocycle morphism WA =wW:A— Io, where the objects vy; of M, for 7 < 7, are inductively determined by the equalities

(CC2): for any two composible arrows k 5 “4. i of I,

(CC3): for any three composible arrows ¢ -> k +, 5 “s iof I,

(CC7): for any arrow a: j — 1 of the category I, the square below commutes.

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