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October 8, 2009

Associativity Data in an (∞,1)-Category

Posted by Mike Shulman

Guest post by Emily Riehl

A popular slogan is that (,1)(\infty,1)-categories (also called quasi-categories or \infty-categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to someone who is happy to regard spaces as Kan complexes, which are simplicial sets in which every horn can be filled. The analogy with categories is somewhat more subtle. We choose to model (,1)(\infty,1)-categories as quasi-categories, which are a particular type of simplicial set. When we regard the 0-simplices of an quasi-category as its objects and the 1-simplices as its morphisms, we can define a weak composition law that is well-defined, unital, and associative only up to homotopy. This means exactly that when we replace the 1-simplices by homotopy classes of 1-simplices we obtain an ordinary category, called the homotopy category of our quasi-category.

However, a lot of data is lost when we replace an quasi-category by its homotopy category. In particular, there exists a 3-simplex that witnesses the fact that a particular composite (hg)f(h g)f of 1-simplices ff, gg, and hh (where some composite hgh g of gg and hh has already been chosen) is also a composite of gfg f (a chosen composite of ff and gg) and hh. This is well-known by those who are familiar with the construction of the homotopy category mentioned above, but a question remains: what sort of associativity data is provided by the nn-simplices, for n>3n \gt 3?

Each nn-simplex of an quasi-category exhibits some composite of the nn morphisms which make up its spine. We will argue that these simplices can be regarded as “unbiased associahedra” in the sense that they witness the “commutativity” of their boundaries, which are n1n-1-simplices of the same form. In fact, one may choose all but one of these n1n-1-simplices to be any witnesses that you want, subject to some obvious constraints. We will describe in detail the combinatorial analogies between these constraints and Stasheff’s associahedra in what follows, at least in low (n6n \leq 6) dimensions.

Before turning to these concrete details, we should acknowledge a more conceptual explanation for the experts in this field, due to Jacob Lurie, for the phenomena we’ll describe below. The theory of quasi-categories is equivalent to the theory of simplicial (or topological) categories. One feature of this equivalence is that the mapping spaces in any quasi-category (however one chooses to extract them) are homotopy equivalent to mapping spaces in a simplicial category, where there is a strictly associative multiplication. Transporting this structure along the homotopy equivalences will give an A A_{\infty} structure on the mapping spaces in the original quasi-category. By definition, A A_{\infty}-spaces are algebras for the non-Σ\Sigma operad {K n}\{ K_n \} with objects given by Stasheff’s associahedra, accounting for their appearance in what follows.

We begin by explaining the construction of the homotopy category of an quasi-category, emphasizing the role played by the 2- and 3-simplices. The reader who already knows this story can comfortably skip to the next section, after making note of our convention for labeling 3-simplices. In the second section, we describe the low-dimensional higher associativity data that is present in any quasi-category, beginning with n=4n=4 and then moving on to n=5n=5 and n=6n=6. A LaTeXed version of this post with even more details can be found here.

Preliminaries, or how is an (,1)(\infty,1)-category “category-like”?

We’ll use standard notation for simplicial sets, which agrees with Lurie’s Higher Topos Theory or his article “What is an \infty-category?” from the Notices of the AMS, although we prefer the name quasi-category for his \infty-category. In particular, we write Δ n\Delta^n for the simplicial set Δ opSet{\mathbf{\Delta}}^{\text{op}} \rightarrow \mathbf{Set} represented by the object [n]={0,1,,n}[n] = \{0,1,\ldots,n\} of Δ{\mathbf{\Delta}}. If S:Δ opSetS : {\mathbf{\Delta}}^{\text{op}} \rightarrow \mathbf{Set} is a generic simplicial set, S nS_n is the set of its nn-simplices. We have face maps d i:S nS n1d_i : S_n \rightarrow S_{n-1} and degeneracy maps s i:S nS n+1s_i : S_n \rightarrow S_{n+1} for 0in0 \leq i \leq n. A horn Λ i n\Lambda^n_i is a simplicial subset of Δ n\Delta^n (or also of its boundary Δ n\partial \Delta^n). It is generated by nn n1n-1-simplices which satisfy the relations typical of the collection of all faces but the iith of any nn-simplex. We are more often interested in relative horns, i.e., maps Λ i nS\Lambda^n_i \rightarrow S. Concretely, these are specified by elements σ 0,,σ i^,σ nS n1\sigma_0, \ldots, \hat{\sigma_i}, \ldots \sigma_n \in S_{n-1} (the images of the generating n1n-1-simplices mentioned previously) satisfying the relations

(1)d jσ k=d k1σ jfor all0j<kn,withj,ki.d_j \sigma_k = d_{k-1} \sigma_j\qquad \text{for all}\; 0 \leq j \lt k \leq n,\; \text{with}\; j,k \neq i.

An quasi-category is a simplicial set SS such that every inner horn can be filled, which means that for every collection of n1n-1-simplices as above with 0<i<n0 \lt i \lt n, there exists σ iS n1\sigma_i \in S_{n-1} and σS n\sigma \in S_n such that the faces of σ\sigma are precisely the σ j\sigma_j.

Fix an quasi-category SS. It is category-like in the following sense. We regard the vertices aka points aka 0-simplices as objects and edges aka arrows aka maps aka 1-simplices fS 1f \in S_1 as morphisms from d 1fd_1f to d 0fd_0f; we write d 1ffd 0fd_1f \stackrel{f}{\rightarrow} d_0f. For any xS 0x \in S_0, s 0xS 1s_0x \in S_1 is a morphism from xx to xx, which we regard as the identity at xx and depict with an equals sign.

We will depict 2-simplices like this

The vertices will always be in the same position, so we typically omit the numbers. The iith edge is the edge opposite vertex ii. We say morphisms ff and ff' with the same endpoints are homotopic if there exists a 2-simplex with boundary

or with ff and ff' switched. In an quasi-category, this defines an equivalence relation and furthermore any one of these conditions implies all of the others.

Importantly, we have a weak composition law defined as follows. Morphisms ff and gg are composable when the codomain d 0fd_0f equals the domain d 1gd_1g, i.e., exactly when

determines a horn Λ 1 2S\Lambda^2_1 \rightarrow S (in words, a 2,1-horn in SS). Any 2-simplex σS 2\sigma \in S_2 that fills this horn exhibits the edge d 1σd_1\sigma as a composite of ff and gg. As SS is an quasi-category, some such σ\sigma always exists. In an quasi-category, any two such composites are homotopic, and furthermore if we replace ff or gg by homotopic maps, the composites are still homotopic.

All of the unproven claims above follow easily by filling 3,1- or 3,2-horns in SS to obtain a 3-simplex. By constructing the horn appropriately, the “missing face” that is given by its filler will be the desired 2-simplex (exhibiting, e.g., that some arrow is a composite of two other arrows, or that some pair of arrows are homotopic). Let’s agree to draw 3-simplices like this

The vertices will always be in the same position, so we typically omit the numbers. Using the conventional numbering for simplices, the 3rd face is the left face, the 2nd face is the back face, the 1st face is the bottom face, and the 0th face is the right face.

We can now prove that composition is associative up to homotopy. Suppose we have three composable morphisms ff, gg, and hh and let us choose composites gfg f and hgh g, as well as 2-simplices exhibiting these chosen composites. We can either choose a composite h(gf)h(g f) and show that it is a composite of ff and hgh g or choose a composite (hg)f(h g)f and show that it is a composite of gfg f and hh. The former strategy amounts to filling the 3,2-horn depicted by

(2)

with the back arrow equal to the chosen morphism h(gf)h(g f). Filling this horn fills in the back face, which exhibits h(gf)h(g f) as a composite of ff and hgh g. The latter strategy amounts to filling the 3,1-horn also depicted by (2) but with the back arrow equal to (hg)f(h g)f. It follows that h(gf)h(g f) and (hg)f(h g)f are homotopic, as they are both composites of the same morphisms.

We have proven that the simplicial set SS is category-like: given SS we can obtain an actual category with objects the 0-simplices and morphisms homotopy classes of 1-simplices. The facts stated above define composition on homotopy classes and prove that it is associative and unital. This category is called the homotopy category of SS and is depicted in the literature by τ 1S\tau_1S (in the unpublished work of André Joyal) or hSS (by Jacob Lurie). However, a lot of data is lost when we forget about SS and work instead with its homotopy category, which is why quasi-categories are objects of interest in their own right.

Higher associativity data

With these preliminaries aside, we are now finally able to address the following question: what sort of higher associativity data is present in any quasi-category? First, a definition from the theory of simplicial sets: the spine of an nn-simplex σS n\sigma \in S_n consists of the edges 010 \rightarrow 1, 121 \rightarrow 2, \ldots , n1nn-1 \rightarrow n between the successive vertices of σ\sigma. In the language of the previous section, the spine of σ\sigma is the maximal list of composable edges of σ\sigma. When n>2n\gt 2, a n,in,i-horn in SS has the same edges as any of its fillers, so we may speak of the spine of a horn as well. For example, in (2), the spine is ff, gg, hh.

Given a sequence ff, gg, hh of three composable morphisms, any 3-simplex with spine ff, gg, hh gives a piece of associativity data associated to that sequence. The faces of any such simplex exhibit composites gfg f and hgh g as well an an arrow d 1fd 0hd_1f \rightarrow d_0h that is simultaneously exhibited as a composite of ff with hgh g and gfg f with hh. The work of the previous section shows that such simplices exist in any quasi-category. This sort of associativity data is more natural than that exhibited, e.g., by a 3-simplex

that fills the 3,1-horn with faces constructed from the faces of (2). This is because the former is unbiased, whereas the latter emphasizes the chosen composite hgh g and neglects to choose a composite of ff and gg.

With this in mind, the next level of associativity data should consist of 4-simplices with spine ff, gg, hh, kk, where this is again a sequence of composable morphisms. We will show firstly that such simplices can be constructed in any quasi-category and secondly that the faces of any such simplex can be understood as some sort of “unbiased associativity pentagon.” We will then explain how the observations made for 4-simplices extend to 5- and 6-simplices, which we explicitly relate to the (rather more complicated) associahedra K 5K_5 and K 6K_6, after which point we become tired of computing associahedra and stop.

4-dimensional associativity

First, we will construct a 4-simplex that should be thought of as a piece of associativity data for the sequence of composable morphisms ff, gg, hh, kk. We will construct this simplex by filling a 4,2-horn. It is also possible to construct such a 4-simplex by filling a 4,1- or 4,3-horn but such constructions feel somewhat less natural. Details for these other cases can be found here.

Any 4-simplex σ\sigma has five faces σ 0,,σ 4\sigma_0, \ldots, \sigma_4, which are themselves 3-simplices. We will construct our 4-simplex by filling the 4,2-horn given by the 3-simplices depicted below, which are themselves constructed by filling 3-dimensional horns:

The labels for the arrows are meant to indicate how these composites are chosen. We begin by choosing 2-simplices exhibiting composites gfg f, hgh g, and khk h of the 3 composable pairs contained in our list of morphisms. Next, we choose simplices exhibiting the composites h(gf)h(g f) and (kh)g(k h)g that will form the 3rd face of σ 1\sigma_1 and 00th face of σ 3\sigma_3, respectively. With the data previously chosen, we can form a 3,1-horn, which we fill to obtain σ 0\sigma_0 and a 3,2-horn, which we fill to obtain σ 4\sigma_4. Finally, we choose a 2-simplex exhibiting a composite (kh)(gf)(k h)(g f) of khk h and gfg f. This allows us to construct a 3,1-horn, which we fill to obtain σ 1\sigma_1 and a 3,2-horn, which we fill to obtain σ 3\sigma_3.

We must show that these 3-simplices fit together to form a 4,2-horn in SS. The conditions that we must check are

(3)d 3σ 4=d 3σ 3gf d 1σ 4=d 3σ 1h(gf) d 0σ 4=d 3σ 0hg d 0σ 3=d 2σ 0(kh)g d 0σ 1=d 0σ 0kh d 1σ 3=d 2σ 1(kh)(gf).\array{ \arrayopts{\colalign{left}} d_3 \sigma_4 = d_3 \sigma_3 \leftrightarrow g f & & d_1 \sigma_4 = d_3 \sigma_1 \leftrightarrow h(g f) \\ d_0 \sigma_4 = d_3 \sigma_0 \leftrightarrow h g& & d_0 \sigma_3 = d_2 \sigma_0 \leftrightarrow (k h)g \\ d_0 \sigma_1 = d_0 \sigma_0 \leftrightarrow k h & & d_1 \sigma_3 = d_2 \sigma_1 \leftrightarrow (k h)(g f).}

Each of these six conditions asks that we have agreed on the 2-simplex that we have chosen to exhibit the indicated composite. E.g., the first condition asks that we have chosen a 2-simplex exhibiting a particular composite of ff and gg. Hence, the above procedure gives rise to a 4,2-horn, which we can fill in SS to obtain the desired 4-simplex σ\sigma.

How can we understand this 4-simplex? First, note that its 5th face is

The faces of σ 2\sigma_2 are exactly the things we haven’t chosen already: 2-simplices exhibiting k(hg)k(h g) as a composite of hgh g and kk, (hg)f(h g)f as a composite of ff and hgh g, and (kh)(gf)(k h)(g f) as a composite of both ff and (kh)g(k h)g and h(gf)h(g f) and kk.

Importantly, the five faces of σ\sigma can be associated with the edges of the associativity pentagon. Using common notation for the latter (such as in the context of a monoidal category), the associations are as follows:

(4)σ 0α g,h,k,σ 1α gf,h,k,σ 2α f,hg,k,σ 3α f,g,kh,σ 4α f,g,h.\sigma_0 \leftrightarrow \alpha_{g,h,k},\; \sigma_1 \leftrightarrow \alpha_{g f,h,k}, \;\sigma_2 \leftrightarrow \alpha_{f,h g,k},\; \sigma_3 \leftrightarrow \alpha_{f,g,k h},\; \sigma_4 \leftrightarrow \alpha_{f,g,h}.

These associations are easily recognized from the spines of each σ i\sigma_i. These five simplices form a simplicial sphere in SS, i.e., a map Δ 4S\partial \Delta^4 \rightarrow S, and σ\sigma fills this sphere. (An nn-dimensional simplicial sphere in SS is specified by n+1n+1 n1n-1-simplices σ 0,,σ n\sigma_0,\ldots, \sigma_n satisfying the relations of (1), except dropping the requirement that j,kij,k \neq i.) In common parlance, a simplicial sphere commutes exactly when it can be filled. So σ\sigma witnesses the fact that the sphere formed by the 3-dimensional associativity data σ 0\sigma_0, σ 1\sigma_1, σ 2\sigma_2, σ 3\sigma_3, σ 4\sigma_4 commutes. In contrast to the situation of the associativity pentagon, this commutativity is unbiased; we don’t have a natural way of composing 3 simplices so we don’t attempt to combine them in any particular order.

5-dimensional associativity

Now we move up a level to consider 5-simplices that can be thought of as a piece of associativity data for the sequence of composable morphisms ff, gg, hh, kk, ll. Any 5-simplex has six faces, which we identify by their spines. Given a 5-simplex σ\sigma with spine ff, gg, hh, kk, ll, its faces have the spines indicated:

(5)σ 0g,h,k,l σ 1gf,h,k,l σ 2f,hg,k,l σ 3f,g,kh,l σ 4f,g,h,lk σ 5f,g,h,k.\array{\arrayopts{\colalign{left}} \sigma_0 \leftrightarrow g, h, k, l & \sigma_1 \leftrightarrow g f, h, k, l & \sigma_2 \leftrightarrow f, h g, k, l \\ \sigma_3 \leftrightarrow f, g, k h, l & \sigma_4 \leftrightarrow f, g, h, l k & \sigma_5 \leftrightarrow f, g, h, k.}

As before it is possible to make a sequence of choices in such a way as to construct a 5,1-, 5,2-, 5,3-, or 5,4-horn whose filler σ\sigma is such an associativity 5-simplex. Rather than describe the details of such a construction, we switch perspectives somewhat and consider together all of the relations satisfied by the six faces that form the simplicial sphere filled by σ\sigma. They are listed in the following table:

(6)d 3σ 5=d 4σ 3 d 2σ 5=d 4σ 2 d 1σ 5=d 4σ 1 d 0σ 4=d 3σ 0 d 0σ 3=d 2σ 0 d 0σ 2=d 1σ 0 d 2σ 4=d 3σ 2 d 1σ 4=d 3σ 1 d 1σ 3=d 2σ 1 d 4σ 5=d 4σ 4 d 0σ 5=d 4σ 0 d 0σ 1=d 0σ 0 d 1σ 2=d 1σ 1 d 2σ 3=d 2σ 2 d 3σ 4=d 3σ 3\array{\arrayopts{\colalign{left}} d_3\sigma_5 = d_4\sigma_3 & d_2 \sigma_5 = d_4 \sigma_2 & d_1 \sigma_5 = d_4 \sigma_1 \\ d_0\sigma_4 = d_3\sigma_0 & d_0 \sigma_3 = d_2 \sigma_0 & d_0 \sigma_2 = d_1 \sigma_0 \\ d_2\sigma_4 = d_3 \sigma_2 & d_1 \sigma_4 = d_3\sigma_1 & d_1 \sigma_3 = d_2 \sigma_1 \\ d_4 \sigma_5 = d_4 \sigma_4 & d_0 \sigma_5 = d_4 \sigma_0 & d_0 \sigma_1 = d_0 \sigma_0 \\ d_1\sigma_2 = d_1\sigma_1 & d_2 \sigma_3 = d_2 \sigma_2 & d_3 \sigma_4 = d_3 \sigma_3 }

These relations ask that certain faces of the 4-simplices σ i\sigma_i are equal, i.e., that we choose certain 3-simplices consistently when constructing the σ i\sigma_i. We’ll identify these chosen 3-simplices with their spines for simplicity. The requirement that two 3-simplices are equal implies that their faces are the same, so we don’t care to specify what these 2-simplices are anymore.

The top three rows of relations correspond, respectively, to the 3-simplices with spines

(7)f,g,kh f,hg,k gf,h,k g,h,lk g,kh,l hg,k,l f,hg,lk gf,h,lk gf,kh,l\array{\arrayopts{\colalign{left}} f,g, k h & & f, h g, k && g f, h, k \\ g, h, l k & & g, k h, l && h g, k, l \\ f, h g, l k & & g f, h, l k & & g f, k h, l }

These correspond to the nine edges of the associahedron K 5K_5

that are shared by two pentagonal faces in the same manner of the correspondence given in the previous section: namely, the 3-simplex with spine x,y,zx,y,z is associated to α x,y,z\alpha_{x,y,z}. This should be compared with (4).

The relations in the fourth row correspond, respectively, to the 3-simplices f,g,hf,g,h; g,h,kg,h,k; and h,k,lh,k,l. We will say more about these relations in a moment. The final row of relations correspond, respectively, to the 3-simplices with spines hgf,k,lh g f, k, l; f,khg,lf, k h g, l; f,g,lkhf,g, l k h. The morphism hgfh g f should be the back edge of the 3-simplex f,g,hf,g,h serving as a common composite for ff with hgh g and gfg f with hh, and similarly of course for the other triples. These last two rows of relations correspond to the edges of the associahedron K 5K_5 that surround the three “naturality squares” in the following manner. For example, the relation d 4σ 5=d 4σ 4d_4\sigma_5 = d_4 \sigma_4 corresponds to the top and bottom arrows of the “naturality square” below; note that these arrows arise from the same natural transformation α f,g,h\alpha_{f,g,h}.

The relation d 1σ 2=d 1σ 1d_1\sigma_2 = d_1\sigma_1 corresponds to the left and right arrows of this square, which arise from the natural transformation α hgf,k,l\alpha_{h g f, k, l}.

The reason these faces of the associahedron are squares and not some other shape has to do with the fact that the information they contain is somehow the product of two K 3K_3, which are intervals geometrically. In the example above, one K 3K_3 (or, extending our analogy downward, one 3-simplex) contains the data associated with the triple f,g,hf,g,h while the other contains the data associated with the triple hgf,k,lh g f, k, l. The product of these two intervals is a square. When we describe the associahedron K 6K_6 in the next section, we will see that it is productive to think of some of its faces as (non-trivial) products in a similar manner.

In conclusion, the 5-simplex σ\sigma can be viewed as data witnessing the commutativity of an unbiased 5-dimensional associahedron whose faces correspond to the lower-dimensional data described above.

6-dimensional associativity and beyond

We extend this analogy one dimension further in part to clarify how the ”naturality squares” of K 5K_5 generalize to higher dimensions. The associahedron K 6K_6 is the 4-dimensional polytope whose vertices correspond to each way of multiplying (non-associatively) an ordered list of six elements. Alternatively, each vertex corresponds to a planar rooted binary tree with six leaves. K 6K_6 has 14 faces, which are themselves polyhedra. Seven of these take the form of the associahedron K 5K_5 while the other seven are pentagonal prisms (geometrically equal to K 4×K 3K_4 \times K_3).

As was the case for lower dimensions, one can construct a 6-simplex with spine f,g,h,k,l,mf, g, h, k, l, m by filling a 6,ii-horn for some 0<i<60 \lt i \lt 6, where the faces of the horn are also constructed by filling horns. As usual, these faces must satisfy the relations (1). We again omit the details of this construction and instead describe the combinatorics of such 6-simplices.

A 6-simplex σ\sigma with spine f,g,h,k,l,mf, g, h, k, l, m will have seven faces corresponding to the seven associahedral faces of K 6K_6. The faces σ 0,,σ 6\sigma_0, \ldots, \sigma_6 satisfy 21 relations, which describe how these 5-simplices are glued together to form the boundary of σ\sigma. Each associahedral face of K 6K_6 is glued along its pentagonal faces to each other face either directly or via a connecting prism, and these gluings correspond bijectively to the 21 relations mentioned above. Additionally, the three squares of an associahedral face are glued to square faces of three separate pentagonal prisms, and some of the pentagonal prisms are glued to each other directly along the remaining square faces. We understand this “lower level gluing” to correspond to the choices that had to be made prior to constructing the 4-simplices that will serve as faces of the σ i\sigma_i. This should be compared with the observation in the previous section that we had to choose a morphism hgfh g f, which should be the back edge of the 3-simplex f,g,hf, g, h before constructing the 3-simplex hgf,k,lh g f, k, l.

The conclusion is the same as we have asserted previously: a 6-simplex σ\sigma can be viewed as data witnessing the commutativity of the unbiased 6-dimensional associahedron whose faces correspond to the lower-dimensional data described above.

Posted at October 8, 2009 9:52 PM UTC

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Re: Associativity data in an (∞,1)-category

This here looks like a typo to me:

Each nn-simplex of an quasi-category exhibits some composite of the nn-morphisms which make up its spine.

The latter “nn” should probably be an “n1n-1”, right?

(I had no luck looking up spine.)

Posted by: Urs Schreiber on October 8, 2009 10:47 PM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

Jim Dolan and I use the term ‘backbone’ instead of ‘spine’, but I think we mean the same thing. An nn-simplex has a bunch of vertices 0,1,,n0,1,\dots, n and an edge from each vertex to each higher-numbered vertex, but these edges:

012n0 \to 1 \to 2 \to \cdots \to n

make up its ‘backbone’.

This makes Emily’s remark make sense (at least to those in the know):

Each nn-simplex of an quasi-category exhibits some composite of the nn-morphisms which make up its spine.

Hmm… well, actually I think she meant “nn morphisms’ rather than ‘nn-morphisms’. The backbone of an nn-simplex consists of nn morphisms, as shown above.

Posted by: John Baez on October 8, 2009 11:00 PM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

Yeah, I figured that after reading further. Was about to secretly change my comment using my superpowers, but you beat me.

At least now spine exists.

Posted by: Urs Schreiber on October 8, 2009 11:05 PM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

If we’re right, I hope Mike changes ‘nn-morphisms’ to ‘nn morphisms’ using his new superpowers!

Posted by: John Baez on October 8, 2009 11:14 PM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

Umm, without using any superpowers, I already see “nn morphisms” in the text, not “nn-morphisms”, and moreover I see the same thing in the original text that Emily sent me.

Posted by: Mike Shulman on October 9, 2009 3:01 AM | Permalink | PGP Sig | Reply to this

Re: Associativity data in an (∞,1)-category

I already see “nn morphisms” in the text, not “nn-morphisms”

Okay, my fault, sorry.

Posted by: Urs Schreiber on October 9, 2009 8:09 AM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

A great guest post, Emily!

Congratulations on your first post and your newfound superpowers, Mike!

People interested in this stuff should take a look at Street’s paper The algebra of oriented simplexes, and the discussion of orientals on the nnLab.

Posted by: John Baez on October 8, 2009 11:22 PM | Permalink | Reply to this

Re: Associativity data in an (∞,1)-category

Sorry for the typo and the confusion it caused. I learned the term “spine” from reading André Joyal’s unpublished treatise on quasi-categories. I wasn’t aware of a source for any of this material, so I had to make up the notation. Suggestions for something more natural would certainly be appreciated.

Posted by: Emily Riehl on October 9, 2009 2:54 AM | Permalink | Reply to this

The product cells

Congratulations, Emily.

This continues a discussion I’ve had with Emily prior to the posting - which posting goes beyond the preliminary version she sent me.

The formal _correspondence_ between product cells and simplicial relations is clear. What I’m still not seeing is the interpretation of the horn fillers as products of (higher) homotopies. That is, in the sense that the filler that defines gf also gives a homotopy between gf and the composite of g and f. A higher dimensional filler gives a higher dimensional homotopy.

Posted by: jim stasheff on October 9, 2009 2:16 PM | Permalink | Reply to this

Re: The product cells

Well, you can choose to regard a 2-simplex from gg and ff to hh as either exhibiting hh as the composite gfg f, or supplying a homotopy from gfg f to hh. In the latter case, you need some other way of giving meaning to “gfg f,” i.e. you need to have some other 2-simplex from gg and ff to kk which you’ve chosen to regard as exhibiting kk as the composite gfg f. Then there’s a 3-dimensional 1-horn whose 3-face is the 2-simplex (f,g,k)(f,g,k), whose 2-face is the 2-simplex (f,g,h)(f,g,h), and whose 0-face is the degenerate 2-simplex (g,id,g)(g,id,g), and a filler of this horn supplies a 2-simplex (k,id,h)(k,id,h) which we can regard as a homotopy from kk (which we have chosen to regard as gfg f) to hh.

Posted by: Mike Shulman on October 9, 2009 9:14 PM | Permalink | PGP Sig | Reply to this

Re: The product cells

Given that there is no canonical composite of ff and gg, I think it seems more natural to regard a 2-simplex as exhibiting some composite hh rather than a homotopy with unspecified domain. As Mike has illustrated, it is possible to find any homotopies you need between two chosen composites by filling a 3-horn. Maybe you can think of the 2-simplex the defines hh as a composite of ff and gg as a piece of data that allows you to find/construct any desired homotopy.

Posted by: Emily Riehl on October 10, 2009 4:34 AM | Permalink | Reply to this

Re: The product cells

There is a setting in which this point can be handled neatly and where both the composite gfgf and the d 1d_1-face of the composition 2-simplex are defined. This is in the homotopy coherent or simplicial nerve of a simplicially enriched category. There the composite gg.ff is the formal composite in the free category on the 1-skeleton, and so hh is there as well. Encorporating the 2-simplices as well allows one to avoid problems of where things are happening and hence the terminology to use.

This dates back to the 1970s with firstly Boardman and Vogt and then Vogt’s lovely paper of homotopy limits. Cordier then formally made the link with the free forget monad on Cat and explicitly defined the homotopy coherent nerve of Top in a paper from about 1982. Cordier and myself then reworked Vogt’s theorem and I like to think that is where some ideas and methods on quasi-categories came to be made more explicit. You might enjoy looking at the three or so papers we wrote on this stuff. Some of this is summarised in the monograph by Heiner Kamps and myself. The references are on the nLab.

On the links with Associahedra and all the ideas that Jim was having at about the same time, he is much more competent than I am to give a `historical’ perspective. There was later on a French thesis (1999) by Florence Marty (I think that was the name) from Montpellier, who looked at a lot of this stuff from a different perspective based on work by Olivier Leroy. She gave a nice talk at one of the Como category theory meetings (probably CT2000). I do not have a copy.

Posted by: Tim Porter on October 10, 2009 7:33 AM | Permalink | Reply to this

Re: The product cells

Tim wrote:

On the links with Associahedra and all the ideas that Jim was having at about the same time, he is much more competent than I am to give a `historical’ perspective.

Not quite the same time - thesis was 1961 and pub date 1963. Rainer Vogt has an excellent history of developments e.g. with Boardman while at Chicago. But I haven’t been able to find it on the web;
his webpage is very user UNfriendly!

Posted by: jim stasheff on October 10, 2009 5:10 PM | Permalink | Reply to this

Re: The product cells

I suspect Jim is referrring to:

`My time as Mike Boardman’s student and our work on infinite loop spaces’ by Rainer. It is in Homotopy Invariant Algebraic Structures, Cont. Math. vol 239 (AMS 1999). There is also a paper there by Jim himself which is worth reading as well.

Posted by: Tim Porter on October 11, 2009 7:58 AM | Permalink | Reply to this

Re: The product cells

Thanks, Tim
For those without access to the book, I’m trying to get a file version from Rainer.

Posted by: jim stasheff on October 12, 2009 2:32 PM | Permalink | Reply to this

Re: The product cells

Ah, that does it! Looking at the 2-simplex two different ways
whihc are not equivalent does explain my confusion.

Posted by: jim stasheff on October 10, 2009 5:06 PM | Permalink | Reply to this

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