S -categories, S -groupoids, Segal categories and quasicategories Timothy Porter The notes were prepared for a series of talks that I gave in Hagen in late * *June and early July 2003, and, with some changes, in the University of La Lagu~na, the Canary * *Islands, in September, 2003. They assume the audience knows some abstract homotopy theory * *and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes * *assume a reasonable knowledge of our book, [26], or any equivalent text if one can be fo* *und! What do the notes set out to do? Ä ims and Objectives!" or should it be "Learning Out- comes"? o To revisit some oldish material on abstract homotopy and simplicially enr* *iched categories, that seems to be being used in today's resurgence of interest in the area* * and to try to view it in a new light, or perhaps from new directions; o To introduce Segal categories and various other tools used by the Nice-To* *ulouse group of abstract homotopy theorists and link them into some of the older ideas; o To introduce Joyal's quasicategories, (previously called weak Kan complex* *es but I agree with Andr'e that his nomenclature is better so will adopt it) and show ho* *w that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cor* *dier and myself; o To ask lots of questions of myself and of the reader. The notes include some material from the `Cubo' article, [35], which was itself* * based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap* * has been kept as small as is feasible as the purpose and the audience of the two sets of note* *s are different and the abstract homotopy theory has `moved on', in part, to try the new methods ou* *t on those same `old' problems and to attack new ones as well. As usual when you try to specify `learning outcomes' you end up asking who * *has done the learning, the audience? Perhaps. The lecturer, most certainly! Acknowledgements 1 I would like to thank Heiner Kamps and his colleagues at the Fern Univerist* *ät for the invitation to give the talks of which these notes are a summary and to the Fern* * Univeristät for the support that made the visit possible, to Jos'e Manuel Garc'ia-Calcines, Jos* *u'e Remedios and their colleagues and for the Departamento de Mathematica Fundamental in the Uni* *versidad de La Laguna, Tenerife, simlarly and also to Carlos Simpson, Bertrand Toen, And* *r'e Joyal, Clemens Berger, Andr'e Hirschowitz and others at the Nice meeting in May 2003, * *since that is where bits of ideas that I had gleaned over a longish period of time fitted tog* *ether so that I think I begin to understand the way that a lot of things interlock in this area* * better than I did before! These notes have also benefitted from comments by Jim Stasheff and some of * *his colleagues on an earlier version. Contents 1 S-categories * * 2 1.1 Categories with simplicial `hom-sets' . . . . . . . . . . . . . . . .* * . . . . . . . . 2 1.2 From simplicial resolutions to S-cats. . . . . . . . . . . . . . . . .* * . . . . . . . . 4 1.3 The Dwyer-Kan `simplicial groupoid' functor. . . . . . . . . . . . . * *. . . . . . . 6 2 Structure * * 8 2.1 The `homotopy' category . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . 8 2.2 Tensoring and Cotensoring . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . 8 3 Nerves and Homotopy Coherent Nerves. 9 3.1 Kan and weak Kan conditions . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . 9 3.2 Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . 11 3.3 Quasi-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . 13 3.4 Homotopy coherent nerves . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . 13 4 Dwyer-Kan Hammock Localisation: more simplicially enriched categories. 17 4.1 Hammocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . 18 4.2 Hammocks in the presence of a calculus of left fractions. . . . . . .* * . . . . . . . 19 5 -spaces, -categories and Segal categories. * * 21 5.1 -spaces, -categories. . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . 23 5.2 Segal categories . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . 24 6 Tamsamani weak n-categories * *25 6.1 Bisimplicial models for a bicategory. . . . . . . . . . . . . . . . .* * . . . . . . . . 26 6.2 Tamsamani-Segal weak n-categories . . . . . . . . . . . . . . . . . . * *. . . . . . . 28 6.3 The Poincar'e weak n-groupoid of Tamsamani . . . . . . . . . . . . . * *. . . . . . 31 7 Conclusion? * * 32 2 1 S-categories 1.1 Categories with simplicial `hom-sets' We assume we have a category A whose objects will be denoted by lower case lett* *er, x,y,z, . . . , at least in the generic case, and for each pair of such objects, (x, y), a simp* *licial set A(x, y) is given; for each triple x, y, z of objects of A, we have a simplicial map, calle* *d composition A(x, y) x A(y, z) -! A(x, z); and for each object x a map [0] ! A(x, x) that `names' or `picks out' the `identity arrow' in the set of 0-simplices of A* *(x, x). This data is to satisfy the obvious axioms, associativity and identity, suitably adapted * *to this situation. Such a set up will be called a simplicially enriched category or more simply an* * S-category. Enriched category theory is a well established branch of category theory. It ha* *s many useful tools and not all of them have yet been exploited for the particular case of S-* *categories and its applications in homotopy theory. Warning: Some authors use the term simplicial category for what we have te* *rmed a simplicially enriched category. There is a close link with the notion of simpli* *cial category that is consistent with usage in simplicial theory per se, since any simplicially en* *riched category can be thought of as a simplicial object in the `category of categories', but a sim* *plicially enriched category is not just a simplicial object in the `category of categories' and no* *t all such simplicial objects correspond to such enriched categories. That being said that usage nee* *d not cause problems provided the reader is aware of the usage in the paper to which refere* *nce is being made. Examples (i) S, the category of simplicial sets: here S(K, L)n := S( [n] x K, L); Composition : for f 2 S(K, L)n, g 2 S(L, M)n, so f : [n] x K ! L, g : [n] x L* * ! M, diagxK [n]xf g g O f := ( [n] x K -! [n] x [n] x K -! [n] x L ! M); ~= Identity : idK : [0] x K ! K, (ii) T op, `the' category of spaces (of course, there are numerous variants* * but you can almost pick whichever one you like as long as the constructions work): T op(X, Y )n := T op( n x X, Y ) Composition and identities are defined analogously to in (i). (iii) For each X, Y 2 Cat, the category of small categories, then we simila* *rly get Cat(X, Y ), Cat(X, Y )n = Cat([n] x X, Y ). 3 We leave the other structure up to the reader. (iv) Crs, the category of crossed complexes: see [26] for background and ot* *her references, and Tonks, [43] for a more detailed treatment of the simplicially enriched cate* *gory structure; Crs(A, B) := Crs(ß(n) C, D) Composition has to be defined using an approximation to the identity, again see* * [43]. (v) Ch+K, the category of positive chain complexes of modules over a commut* *ative ring K. (Details are left to the reader, or follow from the Dold-Kan theorem and exampl* *e (vi) below.) (vi) S(ModK ), the category of simplicial K-modules. The structure uses te* *nsor product with the free simplicial K-module on [n] to define the `hom' and the compositi* *on, so is very much like (i). In general any category of simplicial objects in a `nice enough' category h* *as a simplicial enrichment, although the general argument that gives the construction does not * *always make the structure as transparent as it might be. There is an evident notion of S-enriched functor, so we get a category of `* *small' S-categories, denoted S -Cat. Of course, none of the above examples are `small'. (With regard* * to `small- ness', although sometimes a smallness condition is essential, one can often ign* *ore questions of smallness and, for instance, consider simplicial `sets' where actually the c* *ollections of sim- plices are not truly `sets' (depending on your choice of methods for handling s* *uch foundational questions).) 1.2 From simplicial resolutions to S-cats. The forgetful functor U : Cat ! DGrph0 has a left adjoint, F . Here DGrph0 den* *otes the category of directed graphs with `identity loops', so U forgets just the compos* *ition within each small category but remembers that certain loops are special `identity loops'. T* *he free category functor here takes, between any two objects, all strings of composable non-iden* *tity arrows that start at the first object and end at the second. One can think of F identifying* * the old identity arrow at an object x with the empty string at x. This adjoint pair gives a comonad on Cat in the usual way, and hence a func* *torial simplicial resolution, which we will denote S(A) ! A. In more detail, we write T = F U for* * the functor part of the comonad, the unit of the adjunction j : IdDGrph0! UF gives the comu* *ltiplication F jU : T ! T 2and the counit of the adjuction gives " : F U ! IdCat, that is, "* * : T ! Id. Now for A a small category, set S(A)n = T n+1(A) with face maps di : T n+1(A) ! T n* *(A) given by di= T n-i"T i, and similarly for the degeneracies which use the comultiplicatio* *n in an analogous formula. (Note that there are two conventions possible here. The other will use* * di= T i"T n-i. The only effect of such a change is to reverse the direction of certain `arrows* *' in diagrams later on. The two simplicial structures are `dual' to each other.) This S(A) is a simplicial object in Cat, S(A) : op! Cat, so does not immed* *iately gives us a simplicially enriched category, however its simplicial set of objects is c* *onstant because U and F took note of the identity loops. In more detail, let ob : Cat ! Sets be the functor that picks out the set o* *f objects of a small category, then ob(S(A)) : op! Sets is a constant functor with value the * *set ob(A) of 4 objects of A. More exactly it is a discrete simplicial set, since all its face * *and degeneracy maps are bijections. Using those bijections to identify the possible different sets * *of objects, yields a constant simplicial set where all the face and degeneracy maps are identity map* *s, i.e. we do have a constant simplicial set. Lemma 1 Let B : op ! Cat be a simplicial object in Cat such that ob(B) is a constant s* *implicial set with value B0, say. For each pair (x, y) 2 B0, let B(x, y)n = {oe 2 Bn| dom (oe) = x, codom(oe) = y}, where, of course, dom refers to the domain function in Bn since otherwise dom (* *oe) would have no meaning, similarly for codom. (i) The collection {B(x, y)n| n 2 N} has the structure of a simplicial set * *B(x, y) with face and degeneracies induced from those of B. (ii) The composition in each level of B induces B(x, y) x B(y, z) ! B(x, z). Similarly the identity map in B(x, x) is defined as idx, the identity at x in t* *he category B0. (iii) The resulting structure is an S-enriched category, that will also be * *denoted B. The proof is easy. In particular, this shows that S(A) is a simplicially en* *riched category. The description of the simplices in each dimension of S(A) that start at a and * *end at b is intuitively quite simple. The arrows in the category, T (A) correspond to stri* *ngs of symbols representing non-identity arrows in A itself, those strings being `composable' * *in as much as the domain of the itharrow must be the codomain of the (i - 1)thone and so on. Beca* *use of this we have: S(A)0 consists exactly of such composable chains of maps in A, none of which is* * the identity; S(A)1 consists of such composable chains of maps in A, none of which is the ide* *ntity, together with a choice of bracketting; S(A)2 consists of such composable chains of maps in A, none of which is the ide* *ntity, together with a choice of two levels of bracketting; and so on. Face and degeneracy maps remove or insert brackets, but care must be* * taken when removing innermost brackets as the compositions that can then take place can re* *sult in chains with identities which then need removing, see [8], that is why the comandic des* *cription is so much simpler, as it manages all that itself. To understand S(A) in general it pays to examine the simplest few cases. T* *he key cases are when A = [n], the ordinal {0 < . .<.n} considered as a category in the usua* *l way. The cases n = 0 and n = 1 give no surprises. S[0] has one object 0 and S[0](0, 0 i* *s isomorphic to [0], as the only simplices are degenerate copies of the identity. S[1] like* *wise has a trivial simplicial structure, being just the category [1] considered as an S-category. * * Things do get more interesting at n = 2. The key here is the identification of S[2](0, 2). * *There are two non-degenerate strings or paths that lead from 0 to 2, so S[2](0, 2) will have * *two vertices. The bracketted string ((01)(12)) on removing inner brackets gives (02) and outer br* *ackets, (01)(12) so represents a 1-simplex (01)(12)) (01)(12)______//(02) 5 Other simplicial homs are all [0] or empty. It thus is possible to visualise S* *[2] as a copy of [2] with a 2-cell going towards the bottom: 1@==@ == + == =OEOE 0 __________//_2 The next case n = 3 is even more interesting. S[3](i, j) will be (i) empty if j* * < i, (ii) isomorphic to [0] if i = j or i = j - 1, (iii) isomorphic to [1] by the same reasoning a* *s we just saw for j = i + 2 and that leaves S[3](0, 3). This is a square, [1]2, as follows: ((02)(23)) (02)(23)____________//(03)OO | ss99sOO| | ss | | diagsss | ((01)(12))((23))a| sss b ((01)(13))| | sss | | sss | | ss | (01)(12)(23)((01))((12)(23))//_(01)(13) where the diagonal diag = ((01)(12)(23)), a = (((01)(12))((23))) and b = (((01)* *)((12)(23))). (It is instructive to check that this is correct, firstly because I may have sl* *ipped up (!) as well as seeing the mechanism in action. Removing the innermost brackets is d0, and s* *o on.) The case of S[4] is worth doing. I will not draw the diagrams here although* * aspects of it have implications later, but suggest this as an exercise. As might be expected * *S[4](0, 4) is a cube. Remark The history of this construction is interesting. A variant of it, but with * *topologically enriched categories as the end result, is in the work of Boardman and Vogt, [2] and also* * in Vogt's paper, [44]. Segal's student Leitch used a similar construction to describe a homotopy* * commutative cube (actually a homotopy coherent cube), cf. [29], and this was used by Segal * *in his famous paper, [37], under the name of the `explosion' of A. All this was still in the * *topological framework and the link with the comonad resolution was still not in evidence. Although it* * seems likely that Kan knew of this link between homotopy coherence and the comonadic resolut* *ions by at least 1980, (cf. [15]), the construction does not seem to appear in his work w* *ith Dwyer as being linked with coherence until much later. Cordier made the link explicit in* * [8] and showed how Leitch and Segal's work fitted in to the pattern. His motivation was for th* *e description of homotopy coherent diagrams of topological spaces. Other variants were also appa* *rent in the early work of May on operads, and linked in with Stasheff's work on higher asso* *ciativity and commutativity `up to homotopy'. Cordier and Porter, [9], used an analysis of a locally Kan simplicially enr* *iched category involving this construction to prove a generalisation of Vogt's theorem on cate* *gories of homo- topy coherent diagrams of a given type. (We will return to this aspect a bit l* *ater in these notes, but an elementary introduction to this theory can be found in [26].) Fin* *ally Bill Dwyer, Dan Kan and Justin Smith, [19], introduced a similar construction for an A whic* *h is an S- category to start with, and motivated it by saying that S-functors with domain * *this S-category corresponded to 1-homotopy commutative A-diagrams, yet they do not seem to be a* *ware of 6 the history of the construction, and do not really justify the claim that it do* *es what they say. Their viewpoint is however important as, basically, within the setting of* * Quillen model category structures, this provides a cofibrant replacement construction. Of cou* *rse, any other cofibrant replacement could be substituted for it and so would still allow for * *a description of homotopy coherent diagrams in that context. This important viewpoint can also b* *e traced to Grothendieck's Pursuing Stacks, [22]. The DKS extension of the construction, [19], although simple to do, is ofte* *n useful and so will be outlined next. If A is already a S-category, think of it as a simpl* *icial category, then applying the S-construction to each An will give a bisimplicial category, * *i.e. a functor S(A) : opx op ! Cat. Of this we take the diagonal so the collection of n-sim* *plices is S(A)n,n, and by noticing that the result has a constant simplicial set of objec* *ts, then apply the lemma. 1.3 The Dwyer-Kan `simplicial groupoid' functor. Let K be a simplicial set. Near the start of simplicial homotopy theory, Kan sh* *owed how, if K was reduced (that is, if K0 was a singleton), then the free group functor app* *lied to K in a subtle way, gave a simplicial group whose homotopy groups were those of K, with* * a shift of dimension. With Dwyer in [18], he gave the necessary variant of that constructi* *on to enable it to apply to the non-reduced case. This gives a `simplicial groupoid' G(K) as fo* *llows: The object set of all the groupoids G(K)n will be in bijective corresponden* *ce with the set of vertices K0 of K. Explicitly this object set will be written {__x| x 2 K0}. The groupoid G(K)n is generated by edges __y: ____________d ____________ !1d2.d.d.n+1y0d2.f.d.n+1yory 2 Kn+1 with relations ___s0x= id_______d1d2...dnx. Note since these just `kill' some o* *f the generating edges, the resulting groupoid G(K)n is still a free groupoid. _____ _Define_oei__x=______si+1xfori 0, and, for i > 0, ffii__x= di+1x, but for * *i = 0, ffi0__x= (d1x)(d0x)-1. These definitions yield a simplicial groupoid as is easily checked and, as * *is clear, its simplicial set of objects is constant, so it also can be considered as a simplicially enri* *ched groupoid, G(K). (NB. Beware there are several `typos' in the original paper relating to the* *se formulae for the construction and in some of the related material.) As before it is instructive to compute some examples and we will look at G(* * [2]) and G( [3]. ) simplicially enriched groupoid are free groupoids in each simplicial * *dimension, their structure can be clearly seen from the generating graphs. For instance, G( [2]* *)0 is the free groupoid on the graph __ 1A< 0, and consider the increasing maps ei: [1] ! [p] given by ei(0) = * *i and ei(1) = i+1. For any simplicial set A considered as a functor A : op! Sets, we can evaluate* * A on these ei and, noting that ei(1) = ei+1(0), we get a family of functions Ap ! A1, whic* *h yield a cone diagram, for instance, for p = 3: Ap+UUUUUA +AAA+UUUUA(e1)UUU ++AA UUU + AA UUUUU ++ AA(e2)AA UUU**U + A1 ++ AAA A(e3)+++ AAA d0|| ++ AA__ d1 fflffl| ++ A1 ____//_A0 ++ | ++ |d0 ~~+ fflffl| A1 __d1//_A0 and in general, thus yield a map ffi[p] : Ap ! A1 xA0 A1 xA0 . .x.A0A1. The maps, ffi[p], have been called the Segal maps and will recur throughout the* * rest of these notes. Lemma 3 If A = Ner(C) for some small category C, then for A, the Segal maps are bijecti* *ons. Proof A simplex oe 2 Ner(C)p corresponds uniquely to a composable p-chain of arro* *ws in C, and hence exactly to its image under the relevant Segal map. Better than this is true: Proposition 2 If A is a simplicial set such that the Segal maps are bijections then there is * *a category structure on the directed graph ____//_ A1 ____//_A0oo_. making it a category whose nerve is isomorphic to the given A. Proof To get composition you use ~= d1 A1 xA0 A1 ! A2 ! A1. Associativity is given by A3. The other laws are easy, and illuminating, to che* *ck. The condition `Segal maps are a bijection' is closely related to notions of* * `thinness' as used by Brown and Higgins in the study of crossed complexes and their relationship t* *o !-groupoids, see, for instance, [4], and also to Duskin's `hypergroupoid' condition, [14]. Another result that is sometimes useful, is a refinement of `groupoids give* * Kan complexes':. The proof is `the same': 12 Lemma 4 Let A = Ner(C), the nerve of a category C. (i) Any (n, 0)-horn f : 0[n] ! A for which f(01) is an isomorphism has a filler. Similarly any (n, n)-horn g : * * n[n] ! A for which g(n - 1 n) is an isomorphism, has a filler. (ii) Suppose f is a morphism in C with the property that for any n, any (n,* * 0)-horn OE : 0[n] ! A having f in the (0, 1) position, has a filler, then f is an isomorphi* *sm. (Similarly with (n, 0) replaced by (n, n) with the obvious changes.) Remark Joyal in [25] suggested that the name `weak Kan complex', as introduced by * *Boardman and Vogt, [2], could be changed to that of `quasi-category' to stress the analogy w* *ith categories per se as `Most concepts and results of category theory can be extended to quasi-ca* *tegories', [25]. It would have been nice to have explored Joyal's work on quasi-categories m* *ore fully, e.g. [25], but time did not allow it. The following few sections just skate the surf* *ace of the theory. 3.3 Quasi-categories Categories yield quasi-categories via the nerve construction. Quasi-categories * *yield categories by a `fundamental category' construction that is left adjoint to nerve. This ca* *n be constructed using the free category generated by the 1-skeleton of A, and then factoring ou* *t by a congruence generated by the basic relations : gf h, one for each commuting 1-sphere (g, * *h, f) in A. By a 1-sphere is meant a map a : @ [2] ! A, thus giving three faces, (a0, a1, a2) * *linked in the obvious way. The 1-sphere is said to be commuting if there is a 2-simplex, b 2 * *A2, such that ai= dib for i = 0, 1, 2. This `fundamental category' functor also has a very neat description due to* * Boardman and Vogt. (The treatment here is adapted from [25].) We assume given a quasi-category A. Write gf ~ h if (g, h, f) is a commutin* *g 1-sphere. Let x, y 2 A0 and let A1(x, y) = {f 2 A1 | x = d1f, y = d0f}. If f, g 2 A1(x, y), t* *hen, suggestively writing s0x = 1x, Lemma 5 The four relations f1x ~ g, g1x ~ f, 1yf ~ g and 1yg ~ f are equivalent. The proof is easy. We will say f ' g if any of these is satisfied and call ', the homotopy rel* *ation. It is an equivalence relation on A1(x, y). Set ho A1(x, y) = A1(x, y)= '. If f 2 A1(x, y), g 2 A1(y, z) and h 2 A1(x, z), then the relation gf ~ h in* *duces a map: ho A1(x, y) x ho A1(y, z) ! ho A1(x, z). Proposition 3 The maps ho A1(x, y) x ho A1(y, z) ! ho A1(x, z) give a composition law for a category, ho A, the homotopy category of A. 13 Of course, ho A is the fundamental category of A up to natural isomorphism.* * From previous comments we have: Corollary 1 A quasi-category A is a Kan complex if and only if ho A is a groupoid. 3.4 Homotopy coherent nerves Before introducing this topic, recall some of the intuition behind homotopy coh* *erent (h.c.) diagrams. (Again there is an overview of this theory in [35] and a thorough in* *troduction in [26].) Examples of h.c. diagrams in T op. 1) A diagram indexed by the small category, [2], X(1)G;; X(01)www GGX(12)G www GG ww X(012) G##G X(0) _____X(02)____//X(2) is h.c. if there is specified a homotopy X(012) : I x X(0) ! X(2), X(012) : X(02) ' X(12)X(01). 2) For a diagram indexed by [3]: Draw a 3-simplex, marking the vertices X(0* *), . . . , X(3), the edges X(ij), etc., the faces X(ijk), etc. The homotopies X(ijk) fit togethe* *r to make the sides of a square X(123)X(01) X(13)X(01)O_________//_X(23)X(12)X(01)OOO X(013)|| |X(23)X(012)| | | X(03)_______X(023)__//X(23)X(02) and the diagram is made h.c. by specifying a second level homotopy X(0123) : I2 x X(0) ! X(3) filling this square. These can be continued for larger [n]. Of course, this is not how the theo* *ry is formally specified, but it provides some understanding of the basic idea. The theory was initially developed by Vogt, [44], following methods introdu* *ced with Board- man, [2] (see also the references in that source for other earlier papers on th* *e area). Cordier [8] provides a simple S-category theory way of working with h.c. diagrams and h* *ence released an `arsenal' of categorical tools for working with h.c. diagrams. Some of that * *is worked out in the papers, [10-13] 14 Some Results (i) If X : A ! T opis a commutative diagram and we replace some of the X(a)* * by homotopy equivalent Y (a) with specified homotopy equivalence data: f(a) : X(a) ! Y (a), g(a) : Y (a) ! X(a) H(a) : g(a)f(a) ' Id, K(a) : f(a)g(a) ' Id, then we can combine these data into the construction of a h.c. diagram Y based * *on the objects Y (a) and homotopy coherent maps f : X ! Y, g : Y ! X, etc., making X and Y homotopy equivalent as h.c. diagrams. (This is `really' a result about quasi-categories, see [25].) (ii) Vogt, [44]. If A is a small category, there is a category Coh(A, T op)of h.c. diagrams * *and homotopy classes of h.c. maps between them. Moreover there is an equivalence of categori* *es Coh(A, T op)'! Ho(T opA) This was extended replacing T op by a general locally Kan simplicially enriched* * complete cat- egory, B, in [9]. (iii) Cordier (1980), [8]. Given A, a small category, then the S-category S(A) is such that a h.c. dia* *gram of type A in T opis given precisely by an S-functor F : S(A) ! T op This suggested the extension of h.c. diagrams to other contexts such as a gener* *al locally Kan S-category, B and suggests the definition of homotopy coherent diagram in a S-c* *ategory and thus a h.c. nerve of an S-category. Definition (Cordier (1980), [8], based on earlier ideas of Vogt, and Boardm* *an-Vogt.) Given a simplicially enriched category B, the homotopy coherent nerve of B,* * denoted Nerh.c.(B), is the simplicial `set' with Nerh.c.(B)n = S -Cat(S[n], B). To understand simple h.c. diagrams and thus Nerh.c.(B), we will unpack the * *definition of homotopy coherence. The first thing to note is that for any n and 0 i < j n, S[n](i, j) ~= * * [1]j-i-1, the (j - i - 1)-cube given by the product of j - i - 1 copies of [1]. Thus we can * *reduce the higher homotopy data to being just that, maps from higher dimensional cubes. Next some notation: Given simplicial maps f1 : K1 ! B(x, y), 15 f2 : K2 ! B(y, z), we will denote the composite K1 x K2 ! B(x, y) x B(y, z) !cB(x, z) just by f2.f1 or f2f1. (We already have seen this in the h.c. diagram above f* *or A = [3]. X(123)X(01) is actually X(123)(I x X(01)), whilst X(23)X(012) is exactly what i* *t states.) Suppose now that we have the h.c. diagram F : S(A) ! B. This is an S-functo* *r and so: to each object a of A, it assigns an object F (a) of B; for each string of composable maps in A, oe = (f0, . .,.fn) starting at a and ending at b, a simplicial map F (oe) : S(A)(0, n + 1) ! B(F (a), F (b)), that is, a higher homotopy F (oe) : [1]n ! B(F (a), F (b)), such that (i) if f0 = id, F (oe) = F (@0oe)(proj x [1]n-1) (ii) if fi= id, 0 < i < n F (oe) = F (@ioe(.(Iix m x In-i), where m : I2 ! I is the multiplicative structure on I = [1] by the `max' funct* *ion on {0, 1}; (iii) if fn = id, F (oe) = F (@noe); (iv)iF (oe)|(Ii-1x {0} x In-i) = F (@ioe), 1 i n - 1; (v)i F (oe)|(Ii-1x {1} x In-i) = F (oe0i).F (oei), where oei= (f0, . .,.fi-* *1and oe0= (fi, . .,.fn). We have used @ifor the face operators in the nerve of A. The specification of such a homotopy coherent diagram can be split into two* * parts: (a) specification of certain homotopy coherent simplices, i.e. elements in Nerh* *.c.(B); and (b) specification, via a simplicial mapping from Ner(A) to Nerh.c.(B), of how t* *hese individual parts (from (a)) of the diagram are glued together. The second part of this is easy as it amounts to a simplicial map Ner(A) ! * *Nerh.c.(B), and so we are left with the first part. The following theorem was proved by Cordie* *r and myself, but the idea was essentially in Boardman and Vogt's lecture notes, like so much* * else! Theorem 1 ([9]) If B is a locally Kan S-category then Nerh.c.(B) is a quasi-category. It seems to be the case that if B is only locally weakly Kan, then Nerh.c.(* *B) need not be a quasi-category. 16 The proof of the theorem is in the paper, [9] and is not too complex. The e* *ssential feature is that the very definition (unpacked version) of homotopy coherent diagram makes * *it clear that parts of the data have to be composed together, (recall the composition of simp* *licial maps f1 : K1 ! B(x, y), f2 : K2 ! B(y, z), above and how important that was in the unpacked definition). We thus have that a homotopy coherent diagram `is' a simplicial map F : Ne* *r(A) ! Nerh.c.(B) and that Nerh.c.(B) is a quasi-category. Of course, the usual proof * *that, if X and Y are simplicial sets, and Y is Kan, then S(X, Y ) is Kan as well, extends to hav* *ing Y a quasi- category and the result being a quasi-category. Earlier we referred to Coh(A, B* *) in connection with Vogt's theorem. The neat way of introducing this is as ho S(Ner(A), Nerh.* *c.(B)), the fundamental category of the function quasi-category. In fact, this is essential* *ly the way Vogt first described it. Before we leave homotopy coherence, there is a point that is worth noting f* *or the links with algebraic and categorical models for homotopy types. The S-categories, S[n], co* *ntain a lot of the information needed for the construction of such models. A good example of * *this is the interchange law and its links with Gray categories and Gray groupoids. Consider S[4]. The important information is in the simplicial set S[4](0, 4* *). This is a 3-cube, so is still reasonably easy to visualise. Here it is. The notation is not inten* *ded to be completely consistent with earlier uses but is meant to be more self explanatory. (01)(13)(34)_______________________//_(01)(12)(23)(34)77 oooo OO kk55k OO| oooo || kkkkk | ooo | kkkkkk | | (01)(14)____________________//(01)(12)(24)| | OO | OO | | | | | | | | | || | || | | (03)(34)_____________7_____________//(02)(23)(34)7| | oooo | kk55k | oooo | kkkkk ||oooo || kkkkkk (04)________________________//(02)(24) This looks mysterious! A 4-simplex has 5 vertices, and hence 5 tetrahedral face* *s. Each of the 5 tetrahedral faces will contribute a square to the above diagram, yet a cube has* * 6 square faces! (Things get `worse' in S[5](0, 5), which is a 4-cube, so has 8 cubes as its fac* *es, but [5] has only 6 faces.) Back to the extra face, this is (01)(12)(234) (01)(12)(24)________//_(01)(12)(23)(34)OOOO | | | | (012)(24)|| (012)(234) |(012)(23)(34)| | | | | | | (02)(24)____(02)(234)//_(02)(23)(34). 17 The arrow (012) : (02) ! (01)(12) will, in a homotopy coherent diagram, make it* *s appearence as the homotopy, X(012) : I x X(0) ! X(2), X(012) : X(02) ' X(12)X(01), thus this square implies that the homotopies X(012) and X(234) interact minimal* *ly. Drawing them as 2-cells in the usual way, the square we have above is the interchange s* *quare and the interchange law will hold in this system provided this square is, in some sense* *, commutative. In models for homotopy n-types for n 3, these interchange squares give part o* *f the pairing structure between different levels of the model. They are there in, say, the C* *onduch'e model (2-crossed modules) as the Peiffer lifting, (cf. Conduch'e, [7]) and in the Lod* *ay model, (crossed squares, cf. [30]), as the h-map. In a general dimension, n, there will be pair* *ings like this for any splitting of {0, 1, . .,.n} of the form {0.1. . .,.k} and {k, . .,.n}. 4 Dwyer-Kan Hammock Localisation: more simplicially enriched categories. In his original contribution [36] to abstract homotopy theory, Quillen introduc* *ed the notion of a model category. Such a context is a category, C, together with three classes * *of maps: weak equivalences, W = Cw.e.; fibrations, fib = Cfib; and cofibrations, cofib = Ccof* *ib, satisfying certain axioms so as to give a general framework for `doing homotopy theory'. * *One of the constructions he used was a categorical localisation already well known from Ga* *briel's thesis and the work of the French school of algebraic geometers, (Grothendieck, Verdie* *r, etc.) and concurrently with the publication of [36], studied in some depth by Gabriel and* * Zisman, [20]. The main point was that the analogues of homotopy equivalences, in important in* *stances of homotopical or homological algebra, were only `weak equivalences' so whilst wit* *h a homotopy equivalence between two spaces, you are given two maps, one in each direction, * *plus of course some homotopies, when you have, for instance, a quasi-isomorphism between two c* *hain com- plexes, you only had one map in one direction: f : C ! D together with the know* *ledge that f* : H*(C) ! H*(D) was an isomorphism. The partial solution was to go to the `h* *omotopy category' by formally inverting the weak equivalences, thus getting formal maps* * going in the opposite direction! (This may look like cheating, but really is no worse than i* *ntroducing frac- tions into the integers, so as to be able to solve certain equations, and of co* *urse the detailed construction is closely related!) We thus end up with a category C[W -1]. This construction is very useful, but this homotopy category does not captu* *re the higher order homotopy information implicit in C. For instance, the problem of the `bes* *t' way to handle homotopy limits and colimits, and more generally derived Kan extensions, in a m* *odel category setting is still central to much of the work on abstract homotopy theory, (cf. * *Les D'erivateurs, by George Maltsiniotis, [31] see also [32], Denis-Charles Cisinski's thesis, and s* *ubsequent work, (cf. [5, 6] and related papers), the resum'e of Thomason's note books published by C* *huck Weibel, [46] and Carlos Simpson's, [39], for example). In a series of articles [15-17] * *published in 1980, Dwyer and Kan proposed a neat solution to this problem, simplicial localisation* *s. We will limit ourselves to one of the two versions here, the hammock localisation. 18 4.1 Hammocks Given a category C, and a subcategory W , having the same class of objects, con* *struct a S- category LH (C, W ) or LH C for short, the hammock localisation of C with respe* *ct to W , as follows: The objects of LH C are the same as those of C Given two objects X and Y , the k-simplices of LH C(X, Y ) will be the "red* *uced hammocks of width k and any length" between X and Y . Such a thing is a commutative diag* *ram of form C0,1_______C0,2______. ._.___C0,n-14 ff|iifi | | 444 fi | | | 4 ffiifflffl| fflffl| fflffl|44 fiCfi _______C ______. ._.___C 44 fifi1,1 1,2 1,n-1F44F fifizz|z | | FF 44 ffiizzz| | | FF44F ffiifizzfflffl|z fflffl| fflffl|F44FF X 2D ... ... ... Y 2DD2 xxxx 2DDD2 | | | xxx 22DDD fflffl|| fflffl|| fflffl||xxx 22 22Ck-1,1____Ck-1,2____. ._.___Ck-1,n-1 22 22 | | | 22fflffl|| fflffl|| fflffl|| Ck,1______ Ck,2______. ._.___Ck,n-1 in which (i) the length n of the hammock can be any integer 0, (ii) all the vertical maps are in W , (iii) in each column of horizontal maps, all maps go in the same direction; if * *they go left, then they have to be in W ; plus two reduction conditions, (iv) the maps in adjacent columns go in different directions, and (v) no column contains only identity maps. (If in manipulating hammocks, these last two conditions become violated. th* *en it is simple to reduce the hammock by, for example, composing adjacent columns if they point* * in the same direction or by removing a column of identities. Repeated use of the reductions* * may be needed. One reduction may create a need for another one. It is often useful to work wit* *h unreduced hammocks and then to reduce.) The face and degeneracy maps are defined in the obvious way, (remember the * *vertices of such a simplex are the `zigzags' from X to Y ), however they may result in a no* *n-reduced hammock. Composition is by concatenation followed by reduction: LH C(X, Y ) x LH C(Y, Z) ! LH C(X, Z), expanding the intervening Y node into a vertical line with identities and then * *reducing if need be. Each LH C(X, Y ) is the direct limit of nerves of small categories in an ob* *vious way, i.e. increasing the length n of the hammocks, and so is itself a quasi-category. 19 4.2 Hammocks in the presence of a calculus of left fractions. If the pair (C, W ) satisfies any of the usual `calculus of fractions' type con* *ditions, then the homotopy type of those nerves already stabilises early on in the process (i.e. * *for small n). The argument given in [16] is indirect, so let us briefly see why one of these clai* *ms is true. Suppose that (C, W ) satisfies a calculus of left fractions, then f (i) whenever there is a diagram X0 u X ! Y in C with u 2 W , then there is a* * diagram f0 0v 0 X0! Y Y so that v 2 W and vf = f u, and similarly (ii) if f, g : X ! Y 2 C and u : X ! X0 2 W is such that fu = gu, then there is* * a v 2 W such that vf = vg. (By this means any word in arrows of C and W -1can be rewritten to get all the * *occurrences of arrows from W -1to the left of those `ordinary' arrows from C. Each of the t* *wo substrings can then be composed to reduce the word to one of the form w-1c, i.e. a left f* *raction.) To understand how this reacts with hammocks, consider a simple case where the chos* *en vertex of the hammock LH C(X, Y ) is simply oow__ __c_//_oido_ X C Y Y provide with w 2 W . We construct a new diagram, using the left fractions rule * *(i), giving a 1-simplex with the given vertex at one end: owo__ __c__// oido_ X C Y Y , || | | 0 || || |w |w || || id fflffl|fflffl|||w0 X oo__X_ _c0_//_C0ooY_ so was homotopic to a `left biased' hammock (w0)-1c0. Of course, if the length of the hammock had been greater then the chain of * *`moves' to link it to the `left biased ' form would be longer. Again of course, although combin* *atorially feasible a detailed proof that the left baissed hammocks with vertices of the form X ! C Y provide a deformation retract of LH C(X, Y ) is technically quite messy. Even * *with a better knowledge of what the LH C(X, Y ) looks like, there is still the problem of com* *position. Two left biased hammocks compose by concatenation to give a more general form of ha* *mmock that then gets reduced by the left fractions rules, but these rules do not give a no* *rmal form for the composite. Much as in the composite of arrows in a quasi-category, the composit* *e here is only defined up to homotopy. Suppose we let L1(X, Y ) be the simplicial set of such left biased hammocks* *, then it is a deformation retract of LH C(X, Y ). After composition we reduce to get a diagram L1(X, Y ) x"L1(Y,`Z)______//_TL1(X,"Z)`OO TTTT || '|| TconcatTTTTTT'|reduce| fflffl| TT**T |fflffl| LH C(X, Y ) x LH C(Y, Z)___//_LH C(X, Z) 20 This looks as if it should work well, but if we look at the associativity axiom* *, it is represented by a commutative diagram, and we have replaced each of the nodes of that diagram b* *y a homotopy equivalent object, so we risk getting a homotopy coherent diagram, not a commut* *ative one. This is happening inside LH C, so this does not matter so much. Although attemp* *ting to cut down the size of the `hom-sets' does allow us more control over some aspects of* * the situation, it also has its downside. The solution is to study the homotopy theory of S-categories as such. This* * will lead us back towards the Segal maps as well as continuing to interact with homotopy coh* *erence. For a short time, for the purpose of exposition, we will restrict ourselves* * to small S-categories with a fixed set of objects, O, say, and S-functors will be the identity on obj* *ects. We will denote the category of such things by S -Cat=O. (The material here is adapted * *from [19].) This category has a closed simplicial model category structure in which the sim* *plicial structure is more or less obvious, in which a map D ! D0 is a weak equivalence (resp. a * *fibration), whenever, for every pair of objects, x, y 2 O, the restricted map D(x, y) ! D0(x, y) is a weak equivalence (resp. fibration). (Note, that several of the constructio* *ns we have been looking at gave us weak equivalences in this sense, for instance, S(A) ! A is o* *ne such and that the fibrant objects are the locally Kan S-categories over O). Now as we know any of the categories S-Cat=O form subcategories of the cate* *gory of sim- op plicial categories, Cat . This category also has a closed simplicial model ca* *tegory structure and the nerve and categorical realisation functors induce an equivalence of hom* *otopy categories op (even of the simplicial localisations if you want) between Cat and the categ* *ory of bisimplicial op op sets S . Within Cat we are used to considering S-Cat as a full subcategory* *. Related to the problem of reducing the size of the LH C(X, Y )s is the question of determi* *ning the result of restricting the induced nerve functor to S-Cat. The solution is rather surpr* *ising: op Consider the full subcategory of S determined by those objects X such th* *at (i) X[0] is a discrete simplicial set (cf. the condition on the object simplicial set in an* * S-category); and (ii) for every integer p 2, the Segal map ffi[p] : X[p] ! X[1] xX[0]X[1] xX[0]. .x.X[0]X[1] is a weak equivalence of simplical sets. For reasons that will become clearer later, we will call these objects Sega* *l categories or sometimes Segal 1-categories. Of course, there is a notion of Segal 0-categorie* *s, but these are just nerves of ordinary categories. We will denote the category of these Segal * *1-categories by * * op op Segal-Cat. The result of Dwyer, Kan and Smith, [19], is that the nerve from Cat* * to S , restricts to given an equivalence of homotopy categories between S -Cat and Seg* *al-Cat. In particular this says that any Segal category is weakly equivalent to a bisimpli* *cial set that is a nerve of a simplicially enriched category. Segal categories are weakened simpli* *cial versions of the algebraic structures given by the categorical axioms, so this is in many wa* *ys a coherence theorem for Segal categories rather like the coherence theorems for bicategorie* *s, etc. 21 5 -spaces, -categories and Segal categories. The reason that Segal categories arise as they do is best sought in the paper [* *37] by Segal, although it is not there but rather in [19] that they were introduced, but not * *named as such. (In fact their first naming seems to be in Simpson's [40].) In [37], one of the* * main aims was to get `up-to-homotopy' models for algebraic structures so as to be able to ite* *rate classifying space constructions, to form spectra for studying corresponding cohomology theo* *ries and to help `delooping' spaces where appropriate. Various approaches had been tried, n* *otably that of Boardman and Vogt, [2]. In each case the idea was to mirror the homotopy cohere* *nt algebraic structures that occurred in loop spaces, etc. As an example of the problem, Segal mentions the following: Suppose C is a * *category and that coproducts exist in C. How is this reflected in the nerve of C? It very ne* *arly acquires a composition law, since from X1 and X2, one gets X12= X1 t X2, and two 1-simplic* *es X1 ! X12 X2, but X12 is only determined up to isomorphism. Let C2 be the category of such di* *agrams, i.e. in which the middle is the coproduct of the ends. There is a functor ffi2 : C2 ! C x C and this is an equivalence of categories, but there is also a `composition law' m : C2 ! C given by picking out the coproduct. This looks fine but in fact this tentative* * multiplication again hits the problem of associativity. The theory of monoidal categories was * *not as developed then in 1974 as it is now, and Segal's neat solution was to side-step the issue* *. He formed a category C3 consisting of all diagrams of form X1 _____________//RRX12oo________X2l 22 RRR llllfi 22 RRRR | llll fifi 22 RRRR | lllll fifi 22 RR((Rfflffl|vvll fifi 22 X123<> _b>BB> , fi___ BBffB ff___ BBfiB ___ m BB ___ m BBB __ B__B _ __ b______id_____//_b f(a)____id____//_f(a) which seems like a pretty good version of equivalent objects. The converse is s* *imilar. Tamsamani, [41], generalises this idea to essential surjectivity in higher * *dimensions, and, as Simpson notes in [39], this can be viewed as saying that f : A ! B between two * *T-S weak n-categories is a essentially surjective if T n(f) is surjective. In fact, he p* *oints out that if f is an n-equivalence, then T n(f) is a bijection. Thus we have as a definition that a morphism f : A ! B of T-S weak n-catego* *ries is an n-equivalence if and only if (i) each pair of objects, x, y 2 Ob(A). the induce* *d morphism A1=(x, y) ! B1=(fx, fy) 30 is an (n - 1)-equivalence of weak (n - 1)-categories, and (ii) T n(f) is surjec* *tive. This essentially finishes the definition of T-S weak n-category as we now h* *ave a working definition for all the terms involved. It is amusing and quite useful that T gives a functor from weak n Cat to we* *ak (n - 1) Cat. 6.3 The Poincar'e weak n-groupoid of Tamsamani Tamsamani defines a fundamental Poincar'e n-groupoid for an arbitrary space X a* *nd any n. There are some errors in the published version, since an attempt at an iterativ* *e definition fails since no topology has been specified on the set of simplices involved, however * *this slip is rectified later on and the full approach would seem to give a fairly clear method for def* *ining the gadget. It has to be remembered that the result will be a weak n-groupoid, so in dimens* *ion n = 2 we get a bigroupoid and not a 2-groupoid or a double groupoid with connection. Thi* *s means that the object is rather large. Here we will attempt to describe the bigroupoid by * *taking apart the construction in that case. We initially give the construction in general as it * *is easier to do it that way. Let X be a space and M = (m1, . .,.mn), an object of n. Let M := mn x .* * .x. m1, (note the reversal of order). If 0 k n - 1, set Mn-k = (m1, . .,.mn-k-1) a* *nd M0n-k= (mn-k+1, . .,.mn) and for each 0 i mn-k, let videnote the ithvertex of mn-* *k. Let XM = {f : M ! X |for allk, 0 k n - 1, for alli, 0 i mn-k 0 and for allx 2 Mn-k, x02 Mn-k, f(x, vi, x0) = fi(x0) where fi2 XMn-* *k} Note that if mn-k = 0, then there is one vertex only of the corresponding 0 so* * then XM = XMn-k. This encodes into a subcomplex of the n-simplicial singular complex, th* *e constancy rule that we need for a weak n-category. Homotopy in XM _Let_f, g be two elements of XM . We will say that f and g are homotopic and* * write f ' g or f = __gif there is a fl 2 XM,1_such that ffi0(fl) = f and ffi(fl) = g. Homot* *opy is an equivalence relation and we will denote by X M the set of homotopy classes.__ With the obvious identifications, we now define, n(X)M := X M with the ind* *uced face and degeneracy maps. Theorem 2 [41] The n-simplicial set n(X) is a T-S weak n-groupoid. So what does it look like in dimension 2? The definition of XM has quite a few subcases even with n = 2, so we take * *them one at a time. We have `for all k, 0 k 1': k = 0: for all i, 0 i m2, vi, the ith vertex of m2, and for all x1 2* * m1, one has f(vi, x1) = fi(x1). However this places no restriction on f since there are no * *variables in front of the vi. It merely states that fi= f(vi, _). 31 k = 1: for all i, 0 i m1, vi, the ith vertex of m1, and for all x2 2* * m2, one has f(x2, vi) = fi, but here fi has no variables, it is a constant value. The pictu* *re for M = (1, 1) is of a square with constant values on the vertical sides as in our discussion * *of the bicategory model earlier in these notes. fig. 5 The picture for an element in X(2,1)is similar, a vertical prism, 2 x 1, 2 a* *s base and with `constant' vertical edges, i.e. like the shape we saw earlier in the discussion* * of bicategories, i.e. fig 4 on page 27. The case M = (1, 2) is a horizontal prism with constant ends. To study the homotopy relation, we need to look at XM,1. In particular we look at X1,1,1: there are three cases k = 0, 1, 2. As abov* *e the case k = 0 imposes no condition on the singular multi-prism f, since the rule merely state* *s f(vi, _, _) = fi(_, _) in a sense just defining fi. The condition for k = 1, however implies * *that for all vertices viof m2, f(x3, vi, x1) = gi(x1), i.e. is independent of the variable from m3. Finally for k = 2, the restrictio* *n is that f(x3, x2, vi) is a constant function. This means that a homotopy can be represented as a sing* *ular cube in X in which the left and right vertical faces are constant, the top and bottom a* *re independent of the third direction and the other two faces have no other restriction, i.e. * *the homotopy is precisely a homotopy relative to the boundary of the squares in dimension (1, 1* *). This makes it look as if the Tamsamani bigroupoid is essentially the same as that of Hardi* *e, Kamps and Kieboom, (HKK), [24]. It would be interesting to see if Tamsamani's weak 3-grou* *poid could be adapted, as can this bigroupoid case, to allow for a notion of thinness foll* *owed by a quotient to, say, a Gray groupoid, analogously to the way in which the HKK bigroupoid le* *ads in [23], to a 2-groupoid, or a double groupoid with connections. 7 Conclusion? We have looked at the way in which S-categories, their homotopy coherent form, * *Segal 1- categories, and, perhaps, their iterated form, the T-S weak n-categories, enter* * into the two key areas of abstract homotopy theory. Some of the sources used have been fairly re* *cent, so there is still a lot to do. Here is a list of some questions, some better than others: 1. What is the precise link between the Dwyer-Kan S-groupoid and simplicial* * coherence? What do homotopy coherent simplices in G(K) tell one about the models? Do they* * lead to good descriptions of higher interchange elements analogously to the way in whic* *h maps from S[4] to a S-category produce that interchange square? (The work of Ali Mutlu an* *d myself on higher order Peiffer pairings in simplicial groups may be of relevance here, cf* *, [33, 34].) 2. The Tamsamani method starts with a space X and produces a multi-simplici* *al singular complex. That method could be applied to other types of object, for example, a* * simplicial group, a category, and so on. What do the corresponding Poincar'e weak n-groupo* *ids tell one? 32 (Remember that categories model all homotopy types. They just do it in rather a* * difficult way from the point of view of calculations!) 3. Is it true that the Dwyer-Kan hammock localisation of BA with respect to* * level homotopy equivalences is `closely related to' S(S(A), Nerh.c.(B)) for B locally Kan? If * *so `how close'? A lot of light on this problem has been shed by Vogt in [45] in the topological setti* *ng. The question would seem to be of particular importance given the upsurge of interest in A1 -* *categories resulting from new approaches to quantum deformation. 4. Can one construct a homotopy coherent nerve for a general Segal categor* *y (probably yes) and what are its properties? In general what is the precise relationship * *between quasi categories (as a weakening of categories) and Segal categories (also a weakenin* *g of categories)? (This question is vague, of course, and would lead to many interpretations.) 5. Can one unpack a T-S weak 3-category in a sensible way? Is it sensible t* *o try?. 6. How powerful is the DKS coherence theorem for Segal categories? Can a cl* *ear `stand- alone' proof be given that does not depend on a lot of extra machinery? How con* *structive can it be made? 7. If you take a T-S weak n-category as an n-simplicial set and extract (i)* * its diagonal and (ii) its Artin Mazur codiagonal, what does the structure that results look like* *? Is it related, perhaps just in the groupoid case, to hypercrossed complexes or hypergroupoids,* * [14]. 8. To complete the `Grothendieck programme' of pursuing stacks, (i.e. to co* *nstruct (and study) stacks of models for homotopy n-types and to prove for the locally const* *ant n-stacks, some form of Galois-Poincar'e correspondence theorem between equivalence classe* *s of n-stacks and the corresponding n + 1-type model of the base space or topos), one needs a* * good theory of homotopy coherence with those models. The current attempts by Simpson, Toen * *and others (see papers in the references) go a long way towards such a goal, and in work b* *y others in theoretical physics, similar approaches have been tried in very special cases (* *abelian models, via chain complexes etc.). These latter approaches use a lot of the machinery s* *ketched in these notes. It is probably fair to say that all these approaches suffer from the gu* *lf between the technical nature of the machinery and the simple intuitions behind them. This last question is thus to ask is it possible to give a clear intuitive * *approach, to say, 2- stacks using the Segal-category machinery that does not get bogged down in a te* *chnical morass of Quillen model category theory, an enormous amount of weak infinity category * *theory or similar machinery. This is not to say that approaches using those ideas are no* *t providing a necessary step on the road to understanding the Grothendieck problem, but to as* *k for a simple approach that will aid the geometric intuition. References [1] C. Berger, A Cellular Nerve for Higher Categories, Advances in Maths., 169* * (2002) 118- 175. [2] J. M. Boardman and R. M. Vogt, Homotopy Invariant algebraic structures on * *topological spaces, Lecture Notes in Maths 347, (1973), Springer-Verlag. [3] F. Borceux and G. Janelidze, Galois Theories, Cambridge Studies in Advance* *d Mathe- matics, 72, (2001), Cambridge University Press. 33 [4] R. Brown and P.J. Higgins, Tensor products and homotopies for !-groupoids * *and crossed complexes', J. Pure Applied Algebra, 47 (1987) 1-33. [5] D.-C. Cisinski, Images directes cohomologiques dans les cat'egories de mod* *`eles, preprint Universit'e Paris 7, Septembre 2002. [6] D.-C. Cisinski, Propri'et'es universelles et extensions de Kan d'eriv'ees,* * preprint Universit'e Paris 7, Septembre 2002. [7] D. Conduch'e, Modules Crois'es G'en'eralis'es de Longueur 2, J. Pure Appli* *ed Algebra, 34, (1984), 155-178. [8] J.-M. Cordier, Sur la notion de diagramme homotopiquement coh'erent, Proc.* * 3'eme Col- loque sur les Cat'egories, Amiens (1980), Cahiers de Top. G'eom. Diff., 23* *,(1982) 93 -112. [9] J.-M. Cordier and T. Porter, Vogt's Theorem on Categories of Homotopy Cohe* *rent Dia- grams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65-90. [10] J.-M. Cordier and T. Porter, Maps between homotopy coherent diagrams, Top.* * Appls., 28 (1988) 255-275. [11] J.-M. Cordier and T. Porter, Fibrant diagrams, rectifiactions and a constr* *uction of Loday, J. Pure Applied Algebra, 67 (1990) 111-124. [12] J.-M. Cordier and T. Porter, Categorical Aspects of Equivariant Homotopy, * *Applied Cat- egorical Structures 4 (1996) 195 - 212. [13] J.-M. Cordier and T. Porter, Homotopy coherent category theory, Trans. Ame* *r. Math. Soc., 349 (1997) 1 - 54. [14] J. Duskin, Simplicial Methods and the Interpretation of Triple Cohomology,* * Memoirs A.M.S., Vol. 3, 163, (1975). [15] W. G. Dwyer and D. Kan, Simplicial Localization of Categories, J. Pure App* *lied Algebra, 17 (1980) 267-284. [16] W. G. Dwyer and D. Kan, Calculating Simplicial Localizations, J. Pure Appl* *ied Algebra, 18 (1980) 17-35. [17] W. G. Dwyer and D. Kan, Function Complexes in Homotopical Algebra, Topolog* *y, 19 (1980) 427 -440. [18] W. G. Dwyer and D. Kan, Homotopy Theory and Simplicial Groupoids, Proc. Ko* *nink. Neder. Akad., 87 (1984), 379-389. [19] W. G. Dwyer, D. M. Kan, and J. H. Smith, Homotopy Commutative Diagrams and* * their Realizations, J. Pure Applied Algebra, 57 (1989) 5 - 24. [20] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Ergeb* *. Math. Band 35, (1967), Springer-Verlag. 34 [21] P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Ma* *thematics Vol. 174, (1999), Birkhäuser. [22] A. Grothendieck, (1983), Pursuing stacks, manuscript, 600 + pages. . [23] K.A. Hardie, K.H. Kamps, and R.W. Kieboom, A homotopy 2-groupoid of a Haus* *dorff space, Applied Categorical Structures, 8 (2000), 209-234. [24] K.A. Hardie, K.H. Kamps, and R.W. Kieboom, A homotopy bigroupoid of a topo* *logical space, Applied Categorical Structures, 8 (2001) 311-327 [25] A. Joyal, Quasi-categories and Kan complexes, J. Pure Applied Algebra, 175* * (2002) 207- 222. [26] K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, 1* *997, World Scientific. [27] K. H. Kamps and T. Porter, 2-groupoid enrichments in homotopy theory and a* *lgebra, K-theory, 25 (2002) 373 - 409. [28] F. W. Lawvere, Functorial Semantics of Algebraic Theories, Proc. Nat. Acad* *. Sci. USA 50 (1963) 869 -872.4 [29] R. D. Leitch, The Homotopy Commutative Cube, J. London Math. Soc. (2), 9 (* *1974), 23 - 29. [30] J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pur* *e. Appl. Alge- bra, 24 (1982) 179 - 202. [31] G. Maltsiniotis, Introduction `a la th'eorie des D'erivateur* *s, available at http://www.math.jussieu.fr/~maltsin/ . [32] G. Maltsiniotis, La Th'eorie de l'homotopie de Grothendieck, with two appe* *ndices by D.-C. Cisinski, preprint 332, Institut de Math'ematiques de Jussieu, (Paris 6 & * *Paris 7 / CNRS) June 2002. [33] A. Mutlu and T. Porter, Applications of Peiffer pairings in the Moore comp* *lex of a sim- plicial group, Theory and Applications of Categories, 4, No. 7, (1998) 148* *-173. [34] A. Mutlu and T. Porter, Iterated Peiffer pairings in the Moore complex of * *a simplicial group, Applied Categorical Structures, 9 (2001) 111 - 130. [35] T. Porter, Abstract Homotopy Theory: The Interaction of Category Theory an* *d Homotopy Theory, Cubo, 2002. [36] D. G. Quillen, Homotopical Algebra, Lecture Notes in Maths. vol 43, Spring* *er-Verlag, (1967). [37] G. Segal, Categories and Cohomology Theories, Topology, 13 (1974) 293 - 31* *2. 35 [38] C. Simpson, A closed model structure for N-categories, internal Hom, N-sta* *cks and gen- eralized Seifert-van Kampen, available at: arXiv, alg-geom/9704006. [39] C. Simpson, Limits in n-categories, available at: arXiv, alg-geom/9708010. [40] C. Simpson, Effective generalized Siefert-van Kampen: how to calculate X* *, available at:arXiv, q-alg/9710011. [41] Z. Tamsamani, Sur les notions de n-cat'egorie et n-groupo"ide non strictes* * via des ensembles multi-simpliciaux, K-Theory, 16 (1999) 51-99. [42] B. To"en, Vers une interpr'etation galoisienne de la th'eorie de l'homotop* *ie, Cahiers de Top. G'eom. Diff. cat. (to appear). [43] A. P. Tonks, Theory and applications of crossed complexes, Ph.D. Thesis, U* *niversity of Wales, Bangor, (1993), available from: http://www.informatics,bangor.ac.uk* */public/... /mathematics/research/preprints/93/algtop93.html#93.17 [44] R. M. Vogt, Homotopy limits and colimits, Math. Z., 134 (1973) 11-52. [45] R. M. Vogt, Homotopy homomorphisms and the Hammock Localization, Bol. Soc.* * Mat. Mexicana 37 (1992) 431-448. [46] C. Weibel, Homotopy ends and Thomason model categories, Selecta Math. (N.S* *.) 7 (2001) 533-564. T.Porter, Mathematics Department, School of Informatics, University of Wales Bangor, Bangor, Gwynedd, LL57 1UT, United Kingdom. 36