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April 29, 2010

Understanding the Homotopy Coherent Nerve

Posted by Mike Shulman

(guest post by Emily Riehl)

Simplicial categories, more properly called simplicially enriched categories, provide a model for (,1)(\infty,1)-categories which is related to quasi-categories by means of an adjunction :\mathbb{C}:sSet \leftrightarrowsSet-Cat::\mathbb{N}. This adjunction induces a Quillen equivalence between André Joyal’s model structure on simplicial sets and Julie Bergner’s model structure on simplicial categories. Consequently, this adjunction serves as a primary means of translating between the two worlds, figuring prominently, for example, in the straightening construction, which is the (,1)(\infty,1)-analog of the classical Grothendieck construction of a contravariant Cat-valued pseudofunctor from a categorical fibration.

The main disadvantage to this perspective is that the functor \mathbb{C}, although defined explicitly as a pointwise left Kan extension, is not easily understood – or rather, was not easily understood until a recent paper by Dugger and Spivak. Their “necklace characterization,” detailed below, makes it easy to compute the simplicial categories associated to simplicial sets as well as prove a number of results, at least one of which is rather surprising.

We’ll begin by reviewing the definition of the homotopy coherent nerve \mathbb{N}, which we present, following Dugger and Spivak, by means of a comonad resolution, a perspective we find more intuitive than the usual one. We’ll then turn our attention to the left adjoint \mathbb{C} and explain the details of the necklace characterization. At the conclusion, we’ll give a number of applications, most of which can be found in more detail here.

The homotopy coherent nerve

By familiar abstract nonsense, the adjunction \mathbb{C} \dashv \mathbb{N} is uniquely determined by a functor Δ\rightarrowsSet-Cat, which sends each object [n][n] of Δ \subsetCat to a simplicial category Δ n\mathbb{C}\Delta^n. The simplicial category Δ n\mathbb{C}\Delta^n is a cofibrant replacement of the trivial simplicial category structure on the poset category [n][n] and has the same objects 0, 1, \ldots, nn. Its hom-spaces are typically defined to be nerves of certain poset categories, but we prefer an alternate description.

There is a free-forgetful adjunction F:F :rGrph\leftrightarrowCat:U:U between small categories and reflexive graphs, which are directed graphs equipped with an identity edge at each vertex. This adjunction induces a comonad FUFU on Cat and the resulting comonad resolution defines a simplicial object in Cat

for any (ordinary) small category AA. A simplicial object in Cat gives rise to a simplicial category if and only if all of the functors involved are identities on objects, which is the case here. The nn-simplices of the hom-object from aa and bb are the morphisms from hom-set (FU) n+1A(a,b)(F U)^{n+1}A(a,b).

Unraveling the definition, the 0-simplices of this simplicial category are strings of composable non-identity morphisms in AA; the 1-simplices are strings of morphisms where every morphism appears in exactly one set of parentheses; the 2-simplices are strings where every morphism appears in exactly two sets of parentheses; and so forth. The face maps (FU) kϵ(FU) j(F U)^k \epsilon (F U)^j remove the parentheses that are contained in exactly kk others; FUFUϵF U \cdots F U \epsilon composes the morphisms inside the innermost parentheses. The degeneracy maps F(UF) kη(UF) jUF(U F)^k\eta (U F)^j U double up the parentheses that are contained in exactly kk others; FUFηUF\cdots U F \eta U inserts parentheses around each individual morphism.

When we let AA be the poset category [n][n], the simplicial category arising from this comonad resolution is Δ n\mathbb{C}\Delta^n, as usually defined. For example, let’s compute the hom-object Δ 3(0,3)\mathbb{C}\Delta^3(0,3) for the simplicial category associated to the poset category [3][3] with objects 0, 1, 2, 3 and morphisms as shown

(1)

(2)

Examining the morphisms of FU[3]F U[3] from 0 to 3 we see there are four 0-simplices: ll, hjh j, kfk f, and hgfh g f. Similarly, examining FUFU[3]F U F U[3] reveals nine 1-simplices - (hgf)(h g f), (hg)(f)(h g) (f), (h)(gf)(h) (g f), (hj)(h j), (kf)(k f), (h)(g)(f)(h) (g) (f), (h)(j)(h) (j), (k)(f)(k) (f), and (l)(l) - the last four of which are degenerate. Finally, examining FUFUFU[3]F U F U F U[3] reveals two non-degenerate 2-simplices ((hg)(f))((h g) (f)) and ((h)(gf))((h)(g f)) with boundary as indicated

together with 14 degenerate ones. There are no non-degenerate higher simplices. Hence, Δ 3(0,3)=Δ 1×Δ 1\mathbb{C}\Delta^3(0,3) = \Delta^1 \times \Delta^1.

In general, the hom-object Δ n(i,j)\mathbb{C}\Delta^n(i,j) is empty if i>ji \gt j, a point if i=ji=j, and the cube (Δ 1) ji1(\Delta^1)^{j-i-1} when i<ji \lt j.

The homotopy coherent nerve of a simplicial category CC is the simplicial set with C n=\mathbb{N} C_n = sSet-Cat(Δ n,C)(\mathbb{C}\Delta^n, C). So, for example, the data of a 3-simplex in the homotopy coherent nerve of CC consists of four objects xx, yy, zz, ww of CC; 6 vertices fC(x,y) 0f \in C(x,y)_0, gC(y,z) 0g \in C(y,z)_0, hC(z,w) 0h \in C(z,w)_0, jC(x,z) 0j \in C(x,z)_0, kC(y,w) 0k \in C(y,w)_0, and lC(x,w) 0l \in C(x,w)_0; 1-simplices jgfj \rightarrow g f, khgk \rightarrow h g, lhjl \rightarrow h j, lkfl \rightarrow k f, lhgfl \rightarrow h g f (where the new vertices are defined using the horizontal composition of CC); and two 2-simplices in C(x,w)C(x,w) whose boundary edges are depicted below.

Understanding the hom-spaces of X\mathbb{C}X

Now let’s turn our attention to the left adjoint \mathbb{C}. Let XX be a simplicial set. By definition X= [n]Δ X nΔ n.\mathbb{C} X = \int^{[n] \in \mathbf{\Delta}} \coprod_{X_n}\, \mathbb{C} \Delta^n.

Morally,

where X˜ n\tilde{X}_n denotes the non-degenerate nn-simplices of XX.

The objects of X\mathbb{C} X are the vertices of XX. The simplicial categories Δ 0\mathbb{C}\Delta^0 and Δ 1\mathbb{C}\Delta^1 are the free simplicial categories on the poset categories [0][0] and [1][1] respectively, and the free simplicial category functor is a left adjoint and so commutes with colimits. Hence, if XX is 1-skeletal so that X˜ n=\tilde{X}_n = \emptyset for all n>1n\gt 1, then X\mathbb{C} X is the free simplicial category on the graph with vertex set X 0X_0 and edge set X˜ 1\tilde{X}_1. Concretely, this means that the hom-spaces X(x,y)\mathbb{C} X(x,y) are all discrete simplicial sets containing a vertex for each path of edges from xx to yy in XX.

In general, for each 2-simplex of XX with 2nd face ff, 0th face gg, and 1st face jj, there exists a 1-simplex from the vertex jj to the vertex gfg f in X(x,y)\mathbb{C} X(x,y). Furthermore, for each vertex in some hom-space representing a sequence of paths containing jj, there is a 1-simplex connecting it to the vertex representing the same sequence, except with gfg f in place of jj.

However, the 2-skeleton of XX does not determine the 1-skeleta of the hom-spaces. For example, for each 3-simplex of XX as depicted by (1), there is an edge from ll to hgfh g f in X(x,y)\mathbb{C} X(x,y). In general, there is an edge between the vertices represented by paths p 1p rp_1\ldots p_r and q 1q rq_1 \ldots q_r of edges from xx to yy in XX if and only if each edge pp in the first path that does not appear in the second is replaced by a sequence of nn-edges that appear as the spine of some nn-simplex of XX with pp as its diagonal. Here, the spine of an nn-simplex is the sequence of edges between the adjacent vertices, using the usual ordering of the vertices, and the diagonal is the edge [0,n][0,n] from the initial vertex to the final one.

In this way, each edge of X(x,y)\mathbb{C} X(x,y) corresponds to a necklace Δ n 1Δ n rX\Delta^{n_1} \vee \cdots \vee \Delta^{n_r} \rightarrow X in XX, where Δ nΔ k\Delta^n \vee \Delta^k means that the final vertex of the nn-simplex is identified with the initial vertex of the kk-simplex, and the initial and final vertices of the necklace map to xx and yy respectively. A necklace is comprised of a sequence of beads, the Δ n i\Delta^{n_i} above, that are strung together along the joins, defined to be the union of the initial and final vertices of each bead.

The necklace characterization

By a theorem of Dugger and Spivak, necklaces can be used to characterize the higher dimensional simplices of the hom-spaces X(x,y)\mathbb{C}X(x,y) as well, provided we keep track of additional vertex data.

Theorem (DS) Let XX be a simplicial set with vertices xx and yy. An nn-simplex in X(x,y)\mathbb{C} X(x,y) is uniquely represented by a triple (T,f,T)(T, f, \vec{T}), where TT is a necklace; f:TXf : T \rightarrow X is a map of simplicial sets that sends each bead of TT to a non-degenerate simplex of XX and the endpoints of the necklace to the vertices xx and yy respectively; and T\vec{T} is a flag of sets J T=T 0T 1T 2T n1T n=V TJ_T = T^0 \subset T^1 \subset T^2 \subset \cdots \subset T^{n-1} \subset T^n= V_T of vertices V TV_T of TT, where J TJ_T is the set of joins of TT.

Necklaces f:TXf: T \rightarrow X with the property described above are called totally non-degenerate. Note that the map ff need not be injective. If x=yx = y is a vertex with a non-degenerate edge ee, the map e:Δ 1Xe: \Delta^1 \rightarrow X defines a totally non-degenerate necklace in XX.

Importantly, the face maps d i:X(x,y) nX(x,y) n1d_i : \mathbb{C} X(x,y)_n \rightarrow \mathbb{C} X(x,y)_{n-1} can also be described in the language of flags and necklaces. When 0<i<n0 \lt i \lt n, d i(T,f,T)=(T,f,d iT)d_i (T, f, \vec{T}) = (T,f, d_i\vec{T}) where d iTd_i \vec{T} is the flag of sets J TT i^V TJ_T \subset \cdots \subset \widehat{T^i} \subset \cdots \subset V_T with T iT^i removed from the sequence.

To compute d nd_n, restrict TT to the subnecklace T=T| T n1T' = T\vert_{T^{n-1}} with vertices T n1T^{n-1}. If the triple (T,f| T,d nT)(T', f\vert_{T'}, d_n\vec{T}) is totally non-degenerate, this is the nnth face. Otherwise, there exists a unique quotient (T¯,f¯)(\overline{T'}, \overline{f}) of (T,f| T)(T', f\vert_{T'}) that is totally non-degenerate, and d n(T,f,T)=(T¯,f¯,d nT¯)d_n (T, f, \vec{T}) = (\overline{T'}, \overline{f}, \overline{d_n\vec{T}}), where d nT¯\overline{d_n\vec{T}} is the image of d nTd_n \vec{T} under the surjection TT¯T' \rightarrow \overline{T'}.

(3)

To compute d 0d_0, break up the beads of TT along the vertices in T 1T^1 to form a necklace T TT^{''} \subset T with joins T 1T^1; we call this the T 1T^1-splitting of TT. If (T ,f| T ,d 0T)(T^{''}, f\vert_{T^{''}}, d_0\vec{T}) is totally non-degenerate, this is the 0th face. Otherwise, d 0(T,f,T)=(T ¯,f¯,d 0T¯)d_0(T, f, \vec{T})=(\overline{T^{''}}, \overline{f}, \overline{d_0\vec{T}}), the unique totally non-degenerate quotient, as above.

For example, let’s take X=Δ 3X = \Delta^3 and recompute Δ 3(0,3)\mathbb{C}\Delta^3(0,3). The vertices are necklaces whose set of joins must equal the set of vertices. These necklaces necessarily have the form Δ 1Δ 1\Delta^1 \vee \cdots \vee \Delta^1 and are in bijective correspondence with paths of 1-simplices from 0 to 3, as above.

The 1-simplices of Δ 3(0,3)\mathbb{C}\Delta^3(0,3) are precisely the non-degenerate necklaces of Δ 3\Delta^3 from 0 to 3. They have shapes Δ 3\Delta^3, Δ 1Δ 2\Delta^1 \vee \Delta^2, Δ 2Δ 1\Delta^2 \vee \Delta^1, Δ 2\Delta^2 (mapping to the 2-simplex d 1d^1), Δ 2\Delta^2 (mapping to the 2-simplex d 2d^2), Δ 1Δ 1Δ 1\Delta^1 \vee \Delta^1 \vee \Delta^1, Δ 1Δ 1\Delta^1 \vee \Delta^1, Δ 1Δ 1\Delta^1 \vee \Delta^1, and Δ 1\Delta^1, where these are listed in the same order given above (2).

Non-degenerate 2-simplices are necklaces together with a subset of vertices that strictly contains the set of joins. There are only two such, both with shape Δ 3\Delta^3 and flags {0,3}{0,1,3}[3]\{0,3\} \subset \{0,1,3\} \subset [3] or {0,3}{0,2,3}[3]\{0,3\} \subset \{0,2,3\} \subset [3]. It is a good exercise to compute the boundaries of these two 2-simplices to see which is which.

Applications

To my mind, a main selling point of the necklace characterization is that it makes it easy to compute the simplicial category associated to a generic simplicial set. (Exercise: compute the simplicial category Λ k n\mathbb{C}\Lambda^n_k associated to the (n,k)(n,k)-horn.) Even better, it allows us to easily prove a number of theorems.

The first result, describing the cofibrant replacement of an ordinary category regarded as a simplicial category with discrete hom-objects, might be known already, but I can’t find a reference. (Please tell me if you know one!)

Theorem. For any ordinary category AA, the simplicial category NA\mathbb{C}N A is equal to the simplicial category constructed by the comonad resolution described above (where NN is the ordinary nerve).

Proof: A necklace in a category is uniquely determined by its spine and the set of joins. An nn-simplex in the comonad resolution is a string of morphisms, each contained within (n1)(n-1) sets of parentheses. The morphisms in the string describe the spine and the locations of each level of parentheses defines the corresponding set in the flag of vertex data. \square

When I started thinking about these things, I naively hoped that the simplicial categories associated to quasi-categories might have some fibrancy properties. Using the necklaces, we can prove:

Theorem. If XX is a quasi-category, any horn Λ 1 2X(x,y)\Lambda^2_1 \rightarrow \mathbb{C}X(x,y) has a filler.

Proof: Amalgamate the two necklaces given by the horn, and then extend the resulting simplicial set to a larger necklace containing the two. This will determine the missing face and also the filler.\square

However, for most quasi-categories XX, indeed for all quasi-categories that are not themselves categories and then for many categories as well, there are some 3-dimensional inner horns (and likely also 4-dimensional horns) in some X(x,y)\mathbb{C}X(x,y) which cannot be filled. But in higher dimensions all horns (not necessarily inner) can be filled uniquely, as a consequence of the following Theorem, which really surprised me.

Theorem. For any simplicial set XX, X(x,y)\mathbb{C}X(x,y) is 3-coskeletal.

This says that any sphere Δ n\partial \Delta^n in X(x,y)\mathbb{C}X(x,y) with nn at least four can be filled uniquely. The proof is actually quite easy. The necklace of the unique filler to a sphere also represents each of the inner faces and when n4n \geq 4, the flags of these faces also define the flag of the filler. It remains to check that this filler is compatible with the outer faces, but this can be done by considering the appropriate spherical identity.

When XX is a category, the hom-objects of X\mathbb{C}X are in fact 2-coskeletal. The main point of the proof is that the face map d 0d_0 described in (3) is better behaved for necklaces in categories. Examples can be given to show that these results are all optimal, but I suspect I’ve said quite enough already.

Posted at April 29, 2010 3:03 AM UTC

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4 Comments & 0 Trackbacks

Re: Understanding the homotopy coherent nerve

Thanks. Very interesting, indeed.

I am reading this in a bit of a hurry, since I really have to be looking into something else right now. But this bit here certainly looks very interesting:

For any simplicial set XX, X(x,y)\mathbb{C} X(x,y) is 3-coskeletal.

You write:

[this] really surprised me.

Yes, that surprises me, too! :-)

Maybe one should emphasize at this point to the casual reader that X(x,y)\mathbb{C}X (x,y) is not in general a Kan complex, of course, and that the true hom-\infty-groupoid is the Kan fibrant replacement of this, which gives the true (,1)(\infty,1)-categorical hom-object. These, then, will in general not be coskeletal anymore, hence will not be nn-groupoids for finite nn, in general.

All right, this is all I have time for right now. For the moment I added the link to your writeup to the page relation between quasi-categories and simplicial categories which I had been planning to fill with discussion along the lines of your exposition here, but never got around to yet.

Posted by: Urs Schreiber on April 29, 2010 9:07 AM | Permalink | Reply to this

Re: Understanding the homotopy coherent nerve

For the historical record, the comonad approach was first used by Cordier (see the n-Lab entry on the h.c. nerve for references). That paper was then used by us together to investigate the homotopy coherent nerve in a series of papers in the 1980s. I think, still, that that that approach needs emphasising alongside the other approach using cubes and the left adjoint. Cordier also links this in with earlier work by a student of Segal . Cordier’s paper is well worth looking at. It seems that Kan knew of the connection between this comonad approach and homotopy coherence, from certain terminology that he used in some papers in the 1980s but he never made the link explicit. The construction has been rediscovered several times!!!!

Posted by: Tim Porter on April 29, 2010 5:14 PM | Permalink | Reply to this

Re: Understanding the homotopy coherent nerve

Thanks for sharing a bit of the history. I’ll have a look.

Posted by: Emily Riehl on April 29, 2010 11:49 PM | Permalink | Reply to this

Re: Understanding the homotopy coherent nerve

I was really happy to understand this construction, both the fact that \mathbb{C} on categories (and in particular, on Δ\Delta) coincides with the comonad resolution, and the resulting description of the coherent nerve. The definition of the coherent nerve you sometimes read, by first defining the homset of [n]\mathbb{C}[n] as the nerve of some random poset, always made it seem very opaque and mysterious to me, and something I wanted to avoid working with if at all possible. But seeing this description of it, and working through a few examples as you did above, makes it seem much more obvious and natural. In particular, I feel much happier now about the idea of the coherent nerve as a way to produce quasicategories from simplicial categories (and thereby from model categories).

Posted by: Mike Shulman on April 30, 2010 3:18 PM | Permalink | Reply to this

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