The elementary construction of formal anafunctors

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.


Introduction
Anafunctors were introduced by Makkai [Mak96] as new 1-arrows in the 2-category Cat to talk about category theory in the absence of the axiom of choice. The aim was to make functorial those constructions that are only defined by some universal property, rather than by some specified operation. One also recovers the characterisation of equivalences of categories as essentially surjective, fully faithful 1-arrows. The construction by Bartels [Bar06] of the analogous bicategory Cat ana (S, J), whose 1-arrows are anafunctors, starting from the 2-category Cat(S) of internal categories was extended in [Rob12] to variable full sub-2-categories Cat (S) → Cat(S). The canonical inclusion 2-functor Cat (S) → Cat ana (S, J) was there shown to be a 2-categorical localisation in the sense of Pronk [Pro96] at the fully faithful functors which are locally weakly split in the given pretopology J on S.
In these notes I show that given a 2-category K equipped with a (strict) singleton pretopology J whose elements are fully faithful and co-fully faithful arrows, one can construct an analogue K J of the bicategory Cat ana (S, J). The 1-arrows of K J are formal 2-categorical versions of anafunctors, here dubbed J-fractions. The construction of K J is elementary in the sense of only needing the first-order theory of 2-categories, and the construction is Choice-free. The original 2-category K is a wide and locally full sub-bicategory of K J and the inclusion 2-functor A J : K → K J is a bicategorical localisation; this result uses Pronk's comparison theorem from [Pro96], but it should be possible to prove directly using the construction given here.
The following quote from [Sim06] should be kept in mind when reading the elementary calculations in these notes, as no such details have fully appeared in the literature, let alone at the level of generality here: Nonetheless, it is interesting to note the prevalence of formulations leaving "to the reader" parts of the proofs of details of the localization constructions. . . . Another interesting reference is Pronk's paper on localization of 2-categories [21] 1 , pointed out to me by I. Moerdijk. This paper constructs the localization of a 2-category by a subset of 1-morphisms satisfying a generalization of the right fraction condition. . . . the full set of details for the coherence relations on the level of 2-cells is still too much, so the paper ends with: [21 , p. 302:] "It is left to the reader to verify that the above defined isomorphisms a, l and r are natural in their arguments and satisfy the identity coherence axioms." One pleasant feature of the current approach, at least for the author, is that one could take the opposite 2-category everywhere in the current notes and everything will still work fine, only exchanging pullbacks for pushouts everywhere. In this way, one could also localise suitable 2-categories using cospans, rather than spans, for instance 2-categories whose objects are more algebraic in nature, with q : v → x ff. Then assuming the relevant strict pullbacks exist, there is an equality between the pasted 2-cells, as the source and target 1-arrows all lift u × x u → x through q.
Example 2.8. A more complicated example is the equality of pasted 2-cells in The structure of a site on a 2-category is not a common notion so we need to specify what we mean. There are at least two different ways to describe this in the 1-categorical case, namely using sieves and using pretopologies, and it is not clear a priori that they generalise to the same thing for 2-categories. Our definition will be as follows, as this paper only deals with unary sites.
Definition 2.9. A singleton strict pretopology on a 2-category K is a class J of 1-arrows which contains all identity arrows, is closed under composition and the strict pullback an element of J exists and is again in J. We will assume that specified strict pullbacks are given-rather than merely assuming they exist-and that the pullback of an identity 1-arrow is again an identity 1-arrow.
Since this is the same thing as a singleton pretopology on the 1-category underlying the 2-category, we refrain from placing the prefix '2-' in the name. If one merely asks for existence of pullbacks, then one may use a global axiom of choice to make the pullback of a cover an operation.
Example 2.10. Let K be a 2-category which admits specified strict pullbacks. Then ff is a singleton strict pretopology. This is in some sense a degenerate example. The following is more of interest.
Example 2.11. Let S be a finitely complete category with specified limits and J 0 a singleton pretopology on S. Then we have the 2-categories Cat(S) and Gpd(S) of internal categories and groupoids. Let J denote the class of internal functors in either of those 2-categories whose object component is an arrow in J 0 . Then J is a singleton strict pretopology on both Cat(S) and Gpd(S).
In addition, we need to consider a 2-categorical version somewhat analogous to subcanonicity, and here we cannot avoid involving the 2-arrows. This makes the notion essentially 2-categorical, and not just a structure on the underlying 1-category as is the case for Definition 2.9. 3 Definition 2.12. Given a singleton strict pretopology J, we will call it bi-ff if every j : u → x in J is ff and also co-fully faithful (co-ff): for all g, h : x → y and a : g • j ⇒ h • j there is a unique a : g ⇒ h such that a = 1 j • a.
In this paper we do not need to descend 1-arrows down a cover in the pretopology (which is a consequence of representable presheaves being sheaves), but only 2-arrows, so the weaker notion of ff + co-ff is sufficient. We will not here dwell on how this relates to 2-dimensional sheaf theoryà la Street [Str82].
Example 2.13. Continuing Example 2.11, if we additionally assume the pretopology J 0 consists of regular epimorphisms (hence is a subcanonical pretopology on S), then J is a bi-ff singleton strict pretopology on Cat(S) and on Gpd(S). Indeed, the arrows in this pretopology are also regular epimorphisms, though we do not need this here.
This example partly recovers the examples that were used in [Rob12,§8]; variants on this definition will give all examples from loc. cit.
The main object of study of this paper are 2-categories K with a choice of bi-ff singleton strict pretopology J. We shall just refer to these as 2-sites for brevity, though properly speaking it is a very special case of this notion.
As a consequence of our definition of 2-site, we don't just get descent of 2-arrows along covers, but along maps between covers.
Lemma 2.14. Given a 2-site (K, J), and a diagram v k 2 2 r G G w jx where k, j ∈ J, then r is also co-ff.
The following proof, simplifying the author's, is due to the anonymous referee.
Proof. Recall that r being co-ff means that given a 2-arrow a : f • r ⇒ g • r, there is a unique 2-arrow a : f ⇒ g such that a • id r = a. From our definition of 2-site, j and k are both ff and co-ff. This also implies that the two projection maps pr 1 , pr 2 in the next diagram are in J, hence are both co-ff. Since j • pr 2 = j • r • pr 2 and j is ff, there is a unique lift of id j•pr 2 to an invertible 2-arrow r • pr 1 But pr 2 is co-ff, implying r • pr 1 is co-ff, and since pr 1 is co-ff, then so is r (applying two of the cases of Lemma 2.4 in K op ).

The bicategory of J-fractions
We are aiming to localise a 2-category, and in time-honoured tradition we shall call the arrows in the localised 2-category fractions. Fractions are defined relative to a strict pretopology.
For example, given any 1-arrow f : x → y in K, we have the fraction (id x , f ). In particular, we have for any object a the identity fraction, which is (id x , id x ) Sometimes we will also write the 1-arrow x ← u × x v in such a diagram for emphasis, so that the 0-source object is clear. 4 There are certain maps of fractions which are easier to describe and to compose, and the coherence maps of the bicategory we are going to define all turn out to be examples, so we shall spend some time detailing these. We can compose renaming maps and so get a category K R J (x, y) with objects the fractions from x to y and arrows the renaming maps.
As we shall see, we will also have a category with objects the J-fractions and arrows the maps of fractions, and a functor including K R J (x, y) into this latter category. For now we will be content with giving the definition of the arrow component of this functor, without proving functoriality; namely, a renaming map r from (j, f ) to (k, g) as above is sent to the map ι(r) of fractions specified by the 2-arrow (1) where the 2-arrow on the left is the canonical lift of the identity 2-arrow k • r • pr 1 = k • pr 2 through the ff arrow k, using Lemma 2.5.
Definition 3.4. The identity map 1 : The (vertical) composition of maps of J-fractions proceeds as follows. Given 4 Note that, as presented, this 2-arrow alone does not allow us to reconstruct its (1-)source and (1-)target; we take the source and target as implicitly part of the data (cf the definition of arrow in the category of ZFC-sets). 5 The 'renaming transformations' of [Mak96,§1] in the case when K = Cat are a special case of the notion here.
Makkai requires that r is invertible and ar is the identity 2-arrow.
, consider the 2-arrow t 1 ⊕ t 2 filling the diagram 6 (2) , which we shall call the precomposition of t 1 and t 2 . We need to show that this 2-arrow descends along the arrow But pr 13 is co-ff, and the source and target of t 1 ⊕ t 2 factor − → y for i = 1 and i = 3 respectively. Thus t 1 ⊕ t 2 descends uniquely, and we call this descended 2-arrow t 1 + t 2 (note that + is not a commutative operation!), and it gives a map of J-fractions (u 1 , f 1 ) ⇒ (u 3 , f 3 ).
Remark 3.5. If u 1 × x u 2 × x u 3 → u 1 × x u 3 has a section, then the vertical composition t 1 + t 2 is the whiskering of t 1 ⊕ t 2 on the left with this section.
An example of such a section arises when composing two maps of J-fractions, where one of the maps arises from an inverible renaming map. Here we say a renaming map with data (r, a r ) as in Definition 3.3 is invertible if r is an invertible 1-arrow and a r is an invertible 2-arrow. We record a special case of this as a lemma for future reference.
Hence if a r = id, ι(r) + t is given by the 2-arrow One might be concerned with the bracketing of the triple pullback here; for concreteness we can take (u 1 ×x u 2 )×x u 3 , it would not change the final result if we used the other option.
Further, given an invertible renaming map q from (k, g) to x k ← − v g − → y, with k = k q and g = g q (hence a q = id), the composite t + ι(q) is given by the 2-arrow Proof. Apply the first case in Lemma 3.6 to when t = ι(r ) for an arbitrary renaming transformation q and use Lemma 2.5.
Proposition 3.8. We have a category K J (x, y) with objects the J-fractions from x to y and arrows the maps of J-fractions.
− → y follows from the second and third cases of Lemma 3.6, taking r = id u and q = id v respectively.
We thus need only to show composition is associative. Consider the diagram The bolded objects form a sub-diagram we will refer to below. We will show that the composites c c y a are equal to (3) for a = (t 1 + t 2 ) + t 3 and a = t 1 + (t 2 + t 3 ). First consider (t 1 + t 2 ) + t 3 : ommitting some of the labels on the 1-arrows for clarity. Now the whiskered 2-arrow in the subdiagram on the bold symbols above is equal to the composite 2-arrow in the subdiagram of (3) on the bold symbols, hence the whole diagram equals (3). A symmetric argument shows that t 1 +(t 2 +t 3 )•1 u1234→u14 is also equal to (3). By uniqueness of descent, composition of maps of J-fractions is associative, and K J (x, y) is a category.
3.1. Defining the bicategory K J . Now we want to show that K J (x, y) is the hom-category of a bicategory, so we need a composition functor. Composing 1-arrows is easy: Definition 3.9. The composition of J-fractions is the composite span where recall we are assuming we have specified pullbacks of 1-arrows in J, so this is well-defined.
We shall define the composition in the bicategory K J by defining left and right whiskering functors and proving the interchange law as outlined in [Mak96, pp 126-127] 7 for the case where K = Cat and J is the class of fully faithful, surjective-on-objects functors.
Proof. First, let us show right whiskering preserves identity 2-arrows. That is, the horizontal composition of a pair of identity 2-arrows is the identity 2-arrow of the composition of the 1-arrows. Let x j ← − u f − → y be a fraction and consider the right whiskering of the map id (j,f ) by (l, h). This is the map of fractions given by where the 2-arrow is the unique lift of the unlabelled maps being the obvious projections. But we have the equality Thus whiskering is unital. Now to prove that right whiskering preserves composition we will again use uniqueness of descent, and prove equal a pair of 2-arrows with 0-source a cover of the 0-source of the 2-arrows we are interested in. Without loss of generality, we can right whisker by the fraction (l, id) = y Consider the composable pair of maps of fractions given by the data Let u 123 := u 1 × x u 2 × x u 3 and similarly for u 12 , u 23 , and consider the diagram We need to prove equal the pair of 2-arrows (ρ (l,id) a 1 ) + (ρ (l,id) a 2 ) and ρ (l,id) (a 1 + a 2 ) between the two 1-arrows (w × y u 1 ) × x (u 3 × y w) w × y u 13 × y w pr i −→ w, for i = 1, 3. In Figure 1 the sub-diagram consisting of just the solid arrows together with the 2-arrows between them 2-commutes, so the precomposition (ρ (l,id) a 1 )⊕(ρ (l,id) a 2 ) is given by the top layer of the diagram, namely w ×y u 12 ×y w is given by the unique descent of this 2-arrow along p. The 2-arrow marked ( * ) is the whiskering ρ (l,id) (a 1 + a 2 ), and forms a 2-commuting diagram with a 1 + a 2 and the 1-arrows w × y u 13 × y w → u 13 and w → y. The 2-cell in Figure 1 is, by uniqueness of lifts through p and w → y , which is what we needed to prove.
The definition of the left whiskering is slightly more complicated, as it is such that it doesn't permit us to nearly ignore half of the span as we can for right whiskering. What we shall do is define left whiskering by a general J-fraction 3.1.1. Case I: left whiskering by (id u , f ). Let a be a map of fractions from y ← v 1 g − → z to y ← v 2 h − → z, and f : u → y an arrow in K. The whiskered 2-arrow will be a map of fractions from −−−→ z, and so the desired 2-arrow in K will be of the form Definition 3.12. The left whiskering of the map a of J-fractions by the fraction u 3.1.2. Case II: left whiskering by (j, id u ). We have v i → y = u J-covers for i = 1, 2, and now here where the left, bottom and right arrows are all in J. Thus from Lemma 2.14 we have that the top arrow is co-ff. Notice also that there is a trivial factorisation of pr i : v 12 → v i as v 12 → V 12 Definition 3.13. The left whiskering of the map a by x j ← − u id − → u is given by the 2-arrow λ II (j,id) a in K defined via unique descent along the co-ff arrow v 12 × x V 12 → V 12 by the equation Left whiskering by an arbitrary fraction x j ← − u f − → y will then be the composite of the two (putative) functors given by cases I and II.
The proof that left whiskering preserves (vertical) composition will be deferred to appendix A, as it is a sizable calculation.
Proof. (Left whiskering is unital) We want to do the whiskering Note that without loss of generality we can assume g = id v , the general case follows exactly the same argument merely with g right whiskered onto all the 2-cells involved. We treat case I and case II of the definition of left whiskering separately.
Case I. Note that v 12 in this case is v × y v. The left whiskering of the identity map on , and by Lemma 2.5 this is equal to and this is the identity map on the composite u ← u × y v → v, as required. Case II. Again, in this case, v 12 = v × u v, which for now will be denoted v [2] and V 12 = v × x v. Recall that the 2-cell component of the whiskered identity map will be the unique 2-cell λ : Putting case I and case II together, we have that left whiskering λ (j,f ) : K J (y, z) → K J (x, z) preserves identity maps.
Our unitors consist of identity arrows, so we need to prove that these diagrams commute on the nose. This is true at the level of objects, so we just need to check that the appropriate wiskerings of an arbitrary map of J-fractions (i.e. an arrow in K J (x, y) by identity fractions result in the original map of fractions.
In the case of r x , we can apply Defintion 3.12, whiskering by the fraction (id x , id x ). But then we just get the original map of J-spans. Thus the left triangle above commutes and r x is natural.
In the case of l y , we use Definition 3.10, with y l ← − w h − → z being (id y , id y ). But then w × y,f (u × x v) × g,y w = u × x v, and we are lifting through an identity arrow, and then whiskering (in K) with id y . The result is then the original map of J-spans, making the right triangle above commute and so l y is natural.
Definition 3.17. The associator for the 3-tuple of composable fractions • (j, f ) of J-fractions associated to the renaming map arising from the canonical isomorphism w over x 1 , together the appropriate identity 2-arrow.
Proof. This is proved in Appendix B.
We can check that the associator satisfies the necessary coherence diagrams in the bicategory of fractions and renaming maps, since it will then hold in the bicategory of fractions and maps of fractions. In fact, since the renaming map in question is the associator for products in the strict slice K/x 1 (i.e. strict pullbacks in K), it satisfies coherence by the universal property of pullbacks.
Remark 3.19. If we do not assume that pullbacks of identity arrows are again identity arrows, then we do get nontrivial unitors, but they are, like the associator, renaming maps, and one can check they are coherent.
We have thus proved: 8 Proposition 3.20. There is a bicategory K J with the same objects as K, fractions as 1-arrows and maps of fractions as 2-arrows.
We now define an identity-on-objects strict 2-functor A J : K → K J as follows. For a 1-arrow To check that A J is a strict 2-functor, we need to check first that it is functorial for vertical composition of 2-arrows. In the definition of vertical composition of 2-cells, the diagram (2) in the case of maps of fractions in the image of A J collapses as all objects u i and their fibre products reduce to x, with all arrows between them identity arrows. The descended 2-arrow is then just the vertical composite in K, and so A J preserves vertical composition. It is also simple to show that A J preserves identity 2-arrows. Secondly, we need to show that A J is functorial for horizontal composition. Identity 1-arrows are preserved strictly, as is composition of 1-arrows, so it is just a matter of checking that horizontal composition of 2-cells is preserved. Since horizontal composition is defined via left and right whiskering, we need to check that whiskering a map of fractions in the image of A J by a fraction in the image of A J is of the same form. The right whiskering of A J (a : involves a 2-cell ρ (id,g) a (see Definition 3.10). Since our fractions are in the image of A J , the diagram again collapses so that all appearances of u × x v are equal to x, and w = y, so that ρ (id,g) a = a, and the final result has the 2-cell component the right whiskering of a by g. The left whiskering we need is case I, so we consider Definition 3.12. Consider the map of fractions A J (a : g 1 ⇒ g 2 ) where g 1 , g 2 : y → z (id,f ) a, we have v 12 = v 1 = v 2 = y, the maps between them are identity maps, u = x, and u × y v 12 → v 12 is just f . Thus the whiskered map of fractions is again in the image of A J , and we have proved that A J is a strict 2-functor.
Lemma 3.21. The 2-functor A J is locally fully faithful, that is, K(x, y) → K J (x, y) is fully faithful for all objects x and y of K.
Proof. A map of J-fractions (id x , f ) ⇒ (id x , g) is precisely the same data as a 2-arrow f ⇒ g in K.
Definition 3.22. Given J, a 1-arrow in q : x → y in K is J-locally split if there is an arrow j : u → y in J and a diagram of the form Clearly J ⊂ W J as we are assuming all arrows in J are ff, and every arrow in J is trivially J-locally split.
Proposition 3.23. Let f be a 1-arrow of K. Then A J (f ) an equivalence if and only if f ∈ W J . 8 cf Bartels, who says "The various coherence conditions in a (weak) 2-category are now tedious but straightforward to check." [Bar06] Proof. First assume f : x → y is in J; we will show (id x , f ) is an equivalence, with quasi-inverse (f , id x ). This is because (id x , f ) • (f , id x ) = (f , f ), which is isomorphic to (id y , id y ) by the invertible map of fractions In the other direction, (f , id y )•(id y , f ) = (pr 1 , pr 2 ), where pr i : x× y x → x, i = 1, 2, are the projections (both of which are in J). There is the canonical invertible 2-cell f : pr 1 ⇒ pr 2 , which gives an isomorphism of J-fractions The right hand half of this diagram means that f is J-locally split. Since (id x , f ) is an equivalence it is ff in the bicategory K J . Then as A J is locally fully faithful it reflects ff 1-arrows, hence f is ff in K.
Thus f is both J-locally split and ff, hence is in W J .

For a number of diverse examples of weak equivalences in various categories of internal categories and groupoids, see [Rob12, §8].
3.2. K J as a localisation. Given a 2-category (or bicategory) B with a class W of 1-arrows, we say that a 2-functor Q : B → B is a localisation of B at W if it sends the 1-arrows in W to equivalences in B and is universal with this property. This latter means that for any bicategory A precomposition with Q, is an equivalence of hom-bicategories, with Bicat W meaning the full sub-bicategory on those 2-functors sending arrows in W to equivalences.
Theorem 3.24. A 2-site (K, J) admits a bicategory of fractions for W J , and the inclusion 2-functor A J : K → K J is a localisation at the class W J of weak equivalences.
Proof. That (K, J) admits a bicategory of fractions for W J is [Rob16, Theorem 6] (the weaker hypotheses there on 2-sites are implied by the ones here). The proof that A J is a localisation proceeds via Pronk's comparison theorem [Pro96,Proposition 24], the conditions of which imply that the canonical 2-functor K[W −1 J ] → K J is an equivalence of bicategories. Here K[W −1 J ] is the bicategory of fractions constructed by Pronk, and we recall the conditions of the comparison theorem for ease of reference, using the current notation: EF1. A J is essentially surjective, EF2. For every 1-arrow f of K J there are 1-arrows w ∈ W J and g of K such that A J (g) We now show these conditions hold. To begin with, the 2-functor A J sends weak equivalences to equivalences by Proposition 3.23. EF1. A J is the identity on objects, and hence surjective on objects. EF2. This is equivalent to showing that for any J-fraction x where A J (w) is some pseudoinverse for A J (w). We can take w = j and g = f , since by the proof of Proposition 3.23, (j, id u ) is a pseudoinverse for (id u , j), and the composite fraction of (j, id u ) and (id u , f ) is just (j, f ). EF3. This holds by Lemma 3.21. Thus A J is a localisation of K at W J .
As a last remark, one would like to know if the localisation of K at the weak equivalences is locally essentially small. This can be assured by the following result, where we have used the condition WISC from [Rob12], which states that every object x of K has a set of covers that are weakly initial in the subcategory of K/x on the J-covers.
Proposition 3.25. If the locally essentially small 2-site (K, J) satisfies WISC, then K J is locally essentially small, and hence so is any localisation of K at W J .
Notice that local essential smallness in not automatic, as there are well-pointed toposes with a natural numbers object, otherwise very nice categories, for which the 2-category of internal categories fails the hypothesis of Proposition 3.25. For example the toposes of material sets in models of ZF as given by Gitik (see [vdBM14]) and Karagila [Kar14], or the well-pointed topos of structural sets arising from [Rob15]. Karagila has also described an explicit model of ZF in which the category of anafunctors from the discrete groupoid N to the one-object groupoid B(Z/2) is not essentially small. 9 Finally, note that nothing in this paper relies on K being a (2,1)-category, namely a 2-category with only invertible 2-arrows. This is usually assumed for results subsumed by Theorem 3.24, but is unnecessary in the framework presented here. The following example will be treated in a forthcoming paper.
Example 3.26. Take a pretopos E with stable reflexive coequalisers, and define K to be the wide, locally full sub-2-category of Cat(E) op taking only those internal functors whose object component is a coproduct inclusion, which we shall call a complemented cofibration. Let J consist of the class W of complemented cofibrations f : X → Y that are ff and essentially surjective (i.e. X 0 × f0,Y0,s Y 1 t pr 2 − −− → Y 0 is a regular epimorphism). Then W consists of ff and co-ff arrows, contains all identity arrows, and is closed under composition and-most crucially-pushout. This makes (Cat(E) op , W ) a 2-site as defined in this paper, and the constructions here involving W -fractions in Cat(E) op correspond to analogous dual constructions involving left W -fractions (certain cospans) in Cat(E). The analogous result holds for Gpd(E) in place of Cat(E), and in this case E can be an arbitrary pretopos.
This gives a bicategorical perspective on a generalisation of the case of small groupoids, studied in [LNS17] using cofibration categories as a presentation of (∞, 1)-categories.
Appendix A. Proof that left whiskering in K J preserves vertical composition The definition of left whiskering in K J is slightly more complicated, as it is such that it doesn't permit us to ignore half of the span as we can for right whiskering. Revall that we define left whiskering by a general J-fraction x We now show left whisking by (id, f ) preserves composition. In the following, let λ By uniqueness of descent, λ I (a 1 + a 2 ) = λ I a 1 + λ I a 2 .
A.2. Case II: left whiskering by (j, id u ). Recall the notations V 12 : Definition A.2. The left whiskering of a : (j, g) ⇒ (k, h) by (j, id u ) is given by the 2-arrow λ II (j,id) a in K defined via unique descent by the equation We now prove left whiskering by (j, id u ) preserves composition. In the following, let λ II (−) : (where the subdiagrams on the bold symbols are equal) By uniqueness of descent, we have λ II (a 1 + a 2 ) = λ II a 1 + λ II a 2 . Putting the two results this appendix together, arbitrary left whiskering preserves vertical composition.
Appendix B. Proof that the associator is natural We will use [Mac71, Proposition II.3.2], which says the naturality condition for a transformation between functors A × B × C → D can be checked in each component separately. Thus we only need to show the putative associator for K J is natural for 2-arrows arising from double right whiskering, double left whiskering and left+right whiskering: where these 1-arrows are J-fractions, and the 2-arrows are maps of J-fractions. The major tool here is Lemma 3.6, in particular the second and third cases, since the associators arise from invertible renaming transformations.
For the rest of this appendix, fix composable triples of J-fractions: and maps of fractions a i : (j i , f i ) ⇒ (k i , g i ) for i = 1, 2, 3. Each step will only use one of a 1 , a 2 or a 3 at a time. We also need some notation for pulled back arrows, else there will be a confusing proliferation of projection maps. So, given f , g in the pullback square below (in our given 2-category K), the projection Figure 2. Constructing the double right whiskering maps will be denoted f and  as shown: If we need to pull an arrow g back along two different maps x 1 j1 − → y and x 2 j2 − → y, the results will be denoted g j1 and g j2 respectively. To save space, horizontal composition will be denoted by juxtaposition (in function composition order), and the identity 2-arrow in K on a 1-arrow f will be denoted 1 f . B.1. Double right whiskered. See Definition 3.10 for how right whiskering is defined. We need to both right whisker a 1 twice in succession (by (j 2 , f 2 ) and (j 3 , f 3 )), and also whisker it by (j 3 , f 3 ) • (j 2 , f 2 ) = (j 2 j 3 , f 3 f 2 ). The results then need to be vertically composed with associators in the appropriate order and compared. Namely, we need to show the following 2-arrows are equal: Or rather, we will show (ι(a u1,u2,u3 )) −1 + ρ (j2 3,f3 f2) a 1 + ι(a v1,u2,u3 ) ? = ρ (j3,f3) (ρ (j2,f2) a 1 ).
We thus need to verify that the left whiskerings (in K) of λ (l,h) a and λa with i are equal, since then i being co-ff means λ (l,h) a = λa. But this follows from the definition of the left whiskering of λ (idu,h) a with (l, id u ).
Note that from the diagram we also have (11) pQ = i q P .
B.3. Left+right whiskered. We need to both whisker a 2 by (j 1 , f 1 ) and (j 3 , f 3 ), in that order, and also whisker it by (j 3 , f 3 ) and (j 1 , f 1 ), in that order. The results then need to be vertically composed with associators in the appropriate order and compared. Namely, we need to show the following 2-arrows are equal: .
This completes the proof of Lemma 3.18.
(u 1 u 2 )u 3 Figure 7. Constructing the left then right whiskering Appendix C. Proof that the middle-four interchange holds We need to show the following equality of vertical compositions in K J : (Here the 1-arrows are J-fractions, and the 2-arrows are maps of J-fractions.) For the rest of this appendix, fix composable pairs of J-fractions and maps of fractions a i : (j i , f i ) ⇒ (k i , g i ) for i = 1, 2. In everything that follows, unlabelled 1-arrows are canonical projection maps. We first calculate ρ (j2,f2) a 1 + λ (g1,k1) a 2 . Define the arrows p : v 1 u 2 × x1 v 1 v 2 → u 2 v 2 , i p : v 1 u 2 × v1 v 1 v 2 → v 1 u 2 × x1 v 1 v 2 , and  2 = pr 13 : , and for a, b ∈ {u, v}, the notation a 1 b 2 := a 1 × x2 b 2 . The arrow i p is, by Lemma 2.14, co-ff. Using the definition of right whiskering (Definition 3.10), and left whiskering as in Lemma B.1, we get that the 2-cell representing ρ (j2,f2) a 1 + λ (g1,k1) a 2 is the unique descent along the co-ff arrow u 1 u 2 × x1 (v 1 u 2 × x1 v 1 v 2 ) → u 1 u 2 × x1 v 1 v 2 (using Lemma 2.14) of (21)