equivalences in/of $(\infty,1)$-categories
The homotopy coherent nerve (also called simplicial nerve) of a simplicially enriched category is a simplicial set which includes information about all the higher homotopies present in the hom-spaces. It generalizes the ordinary nerve of an ordinary category.
The homotopy coherent nerve operation
is induced, by the general machinery of nerve and realization, by a cosimplicial simplicially enriched category, namely a functor
from the simplex category to the category of simplicially enriched categories which regards each $n$-simplex as a SSet-enriched category with $n$ objects analogous to how the orientals regard the $n$-simplex as an n-category.
We here describe the cosimplicial sSet-enriched category
that induces the homotopy coherent nerve.
Recall that a reflexive graph is a simplicial set of dimension $1$, i.e. 1-coskeletal; they form a full subcategory $reflGraph\hookrightarrow Cat$. The forgetful functor $U: Cat \to reflGraph$ has a left adjoint $F$ hence $G = FU : Cat\to Cat$ is a comonad. By the definition its cobar construction is an augmented simplicial endofunctor $S\to Id$ featuring $S :\Delta\to sSet Cat$ and whose augmentation is a cofibrant replacement of a 1-category in the Bergner model structure on $sSet$ (“model structure for simplicially enriched categories”).
For $[n]$ the finite ordinal $[n] := \{0 \lt 1 \lt \cdots \lt n\}$ and for $\Delta[n]$ be standard simplicial $n$-simplex, define the $sSet$-category $S[n]$ as follows:
the objects of $S[n]$ are $\{0,1, \cdots, n\}$;
the hom-objects $S[n]_{i,j} \in sSet$ for $i, j \in \{0,1,\cdots, n\}$ are the nerves
of the poset $P_{i,j}$ which is equivalently
the poset of subsets of $[i,j]$ that contain both $i$ and $j$ (so in particular if $i\gt j$ then $P(i,j)$ is empty and hence so is its nerve) with the partial order is given by inclusion.
the poset of paths in $[n]$ that start at $i$ and finish at $j$ (hence is empty if $i\gt j$), the order relation is given by ‘subdivision’, i.e. path $a$ is less than path $b$ in $P(i,j)$ if $b$ visits all the vertices that $i$ does … and perhaps some others as well.
Of course, the way you go between the two descriptions is that a path corresponds to the set of vertices it visits and vice versa.
Notice that the simplicial set $N(P_{i,j})$ is isomorphic to the $j-i-1$ cube in $sSet$:
Under this isomorphism for instance the vertex $(0,0,1,0,1) \in (\Delta[1])^{\times (j-i-1)}$ corresponds to the subset $\{i+3,i+5\} \subset [i,j]$ and to the path $i \to i+3 \to i+5 \to j=i+6$.
(We will look at an example after this definition.)
the composition operation on hom-objects
is induced by ‘concatenation of the corresponding paths’ and thus essentially by union of the sets involved.
The homotopy coherent nerve functor
is the nerve defined by the cosimplicial $sSet$-category $S : \Delta \to sSet Cat$ defined above.
For $C \in sSet Cat$ a simplicially enriched category, the homotopy coherent nerve $N(C)$ is the simplicial set uniquely characterized by the formula
By the general logic of nerve and realization, this functor has a left adjoint
the realization functor given by the coend formula
This functor does extend the functor $S : \Delta \to sSet Cat$ in that there is a canonical isomorphism
and hence may consistently be named $S$.
We illustrate here the nature of the cosimplicial $sSet$-category $S : [n] \mapsto S[n]$.
We will examine the lowest dimensional cases.
For $n = 0$ there is nothing of note.
For $n = 1$ we have that
is the poset with a single object.
For $n = 2$, there are unique paths in $[2]$ from $[0]$ to $[1]$, and $[1]$ to $[2]$, so the corresponding homs in $S[2]$ are copies of $\Delta[0]$ (or, if you prefer, of $\Delta[1]^0$!). Things are slightly more interesting for $S[2](0,2)$. Looking at this from the ‘subsets’ viewpoint, as above, there clearly are two subsets of $\{0,1,2\}$ containing both $0$ and $2$, one corresponds to the direct route in $[2]$ from $0$ to $2$, the other goes via $1$ so is $0\to 1\to 2$.
So in $S[2](0,2)$, there is a 1-simplex $k$ starting at $\{0,2\}$ and ending at $\{0,1,2\}$.
Everything else, in higher dimensions, is degenerate, so $S[2](0,2)\cong \Delta[1]$. Sometimes it is useful to think of this 1-simplex as ‘rewriting’ the direct path to that via 1, all this happening in the free category on the underlying graph of the poset $[2]$. (The construction of $S[n]$ in general has a nice interpretation in terms of higher dimensional rewriting. This can be given using the language of polygraphs or computads.)
In this example there are no significant compositions. To see examples of those, you need to look at $n = 3$. In $S[3]$, the simplicial hom-sets $S[3](i,j)$ for $(i,j) \neq (0,3)$, can all be analysed by the same sort of argument to the above. The new features occur in $S[3](0,3)$. The vertices of this simplicial set are the subsets corresponding to the direct path $0\to 3$ and then the three others. Rewriting the direct path can be done in two immediate ways, to go via the left or via the right route. Each of these can be ‘rewritten’ to give the longest path / largest subset. There is also, of course, an inclusion of the smallest to the largest of these, so that in total the poset here looks like:
In addition, there will be 2-simplexes filling the two triangles, coming from the chains $\{0,3\}\subset \{0,1,3\}\subset \{0,1,2,3\}$ and $\{0,3\}\subset \{0,2,3\}\subset \{0,1,2,3\}$ in the poset.
We thus get $S[3](0,3) \cong \Delta[1]^2$, a square.
The composition maps
and similarly for the one with 1 replaced by 2, are now fairly obvious.
For $n = 4$, the corresponding diagram for $S[4](0,4)$ gives a cube but here there is an interesting feature.
Five of the six faces of the cube $|N(P_{0,4})|$ correspond to the the associativity of composition of triples of composable morphisms in $[4]$. These correspond to the 5 faces of the 4-simplex $\Delta[4]$, as depicted for instance at oriental and at monoidal category.
But the cube has one more face
which does not correspond to associativity: instead, this encodes the exchange law
or, if preferred, to the fact that
is to be a simplicial map.
A similar phenomenon occurs in higher dimensions. There are two ‘extra faces’ in $S[5](0,5)$, and so on.
Any 2-category gives a simplcially enriched category using the embedding of Cat into sSet via the usual nerve functor. The homotopy coherent nerve of a 2-category consideed in this way is, sometimes, called the geometric nerve? of the 2-category. The Duskin nerve of a bicategory is an extension of this construction.
A particular case of this nerve is the nerve of a 2-group considered as a 2-category.
If $C \in sSet Cat$ is such that all hom-objects $C(x,y) \in sSet$ are Kan complexes, then the homotopy coherent nerve $N(C)$ is a weak Kan complex/quasi-category.
If $f : C \to D$ is a morphism of such Kan-complex enriched categories which is a weak equivalence (in the model structure on sSet-categories) in that
the induced functor
on the ordinary underlying homotopy categories (obtained by taking hom-wise connected component sets) is essentially surjective
its component on each hom-object
is a homotopy equivalence,
then its homotopy coherent nerve
is a weak equivalence of quasi-categories.
We may think of category $\Delta[n]$ trivially as a simplicially enriched category. In the model structure on sSet-categories the object $S[n]$ is a cofibrant replacement of $\Delta[n]$. And Kan-complex enriched categories are fibrant. So on these the homotopy coherent nerve is given by the derived hom-space functor
The use of $S[A]$, above, extends that given at the start of this page. Here $S$ is related to the left adjoint of the homotopy coherent nerve, but is defined using a comonadic resolution?. The comonad comes from the adjunction between small categories and directed graphs with distinguished ‘unit’ loops. The ‘forgetful’ part of the adjunction forgets the composition in the category, but remembers that the identity arrows are special. The left adjoint / ‘free’ part of the adjunction takes a directed graph (with distinguished ‘identity’ loops, and forms the free category on the non-identity arrows. As usual, we can form a comonad from this and hence form a functorial simplicial resolution of any small category, $A$.
This can also be seen to be a case of a bar resolution? construction, related to the bar construction. Here the adjoint pair also give a monad on directed graphs with distinguished ‘unit’ loops and the small category $A$ is an algebra for this monad.
Since the functors involved preserve the identities on the objects of $A$, the resulting simplicial category is a simplicially enriched category, and this is $S[A]$. The $n$-dimensional arrows between objects, $a$ and $b$ in $S[A]$ correspond to a path from $a$ to $b$ in $A$ containing no identity arrows, together with a bracketting of the resulting string having depth $n$.
By hom-wise precomposition with the singular complex functor
which is a monoidal functor, the homotopy coherent nerve extends to a nerve of Top-categories
As such, it is a special case of the Boardman-Vogt W-construction for cofibrant replacement of topological operads. See also dendroidal homotopy coherent nerve.
In this construction, rouhgly, for $T$ a tree in an operad $O$, the tree is replaced with the topological space $e(T) \to [0,1]$ of maps from the set of edges of $T$ to the topological unit interval.
We may restrict this construction to the $n$-simplex $\Delta[n]$, regarded as a category and then trivially regarded as a $Top$-category. Then a tree in $\Delta[n]$ is necessarily a linear tree $\to \to \cdots \to$ of some length $k$ and is hence mapped to the topological space of functions $k \to [0,1]$, i.e. to the space $[0,1]^k$. This is the geometric realization of the simplicial cubes $(\Delta[1])^k$ that we saw above.
As mentioned above, the simplicial or h.c. nerve, together with its left adjoint, serves to relate the two models of (∞,1)-categories given by quasi-categories and simplicially enriched categories.
The homotopy coherent nerve extends to a Quillen equivalence between the Joyal model structure $SSet_{Joyal}$ that models quasi-categories and the model structure on SSet-categories.
See
for details.
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).
general pattern | ||||
---|---|---|---|---|
strict enrichment | (∞,1)-category/(∞,1)-operad | |||
$\downarrow$ | $\downarrow$ | |||
enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | ||
(∞,1)Cat | ||||
SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | RelativeSimplicialSets | |
$\downarrow$simplicial nerve | $\downarrow$ | |||
SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||
(∞,1)Operad | ||||
SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |
$\downarrow$dendroidal nerve | $\downarrow$ | |||
SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||
$\mathcal{O}$Mon(∞,1)Cat | ||||
DendroidalCartesianFibrations |
homotopy coherent nerve
The original motivation for the introduction of the homotopy coherent nerve is that it provides a neat simplicial formulation of idea of homotopy coherent diagrams. These were studied in the 1970s, by Boardman and Vogt in joint work, and Vogt individually, and Cordier (reference below).
Cordier realised that, with a slight modification in the definition, Vogt’s definition of homotopy coherent diagram, indexed by a small category $A$, say, corresponded exactly to a simplicially enriched functor from the $SSet$-category $S[A]$ to the $SSet$-category $Top$. They thus also corresponded to simplicial maps from the nerve of $A$ to $N(Top)$, (although that latter object was ‘too large’ to be a simplicial ‘set’). This allowed a good definition of homotopy coherent diagrams in arbitrary simplicially enriched categories to be given.
This definition works best when the simplicially enriched category is ‘locally Kan’, in other words it is enriched in the category of Kan complexes. These locally Kan $SSet$-categories are the fibrant ones in the model structure on sSet-categories.
Cordier and Porter (1986) proved that if $C$ is a locally Kan simplicially enriched category then $N(C)$ is a ‘weak Kan complex’, in other words, a quasi-category. Many of the ideas behind this result can be traced to Vogt’s paper of 1973.
In more modern terminology as Kan complexes can be considered as ∞-groupoids, these locally Kan simplicially enriched categories are one particularly nice model for a (infinity,1)-category, and so this result is one of the earliest giving the transition from one model for (infinity,1)-categorys to another, the ‘weak Kan complexes’ or quasi-categories.
The homotopy coherent nerve operation was introduced, explicitly, in
Cordier made the link with earlier work by R.D. Leitch.
as well as the paper by Vogt (see below) and earlier work of Boardman and Vogt,
With Tim Porter, Cordier proved the simplicial generalisation of a theorem of Vogt in
This theorem describes an equivalence between the category obtained by inverting the ‘levelwise’ homotopy equivalence in a category of diagrams, and the homotopy category of homotopy coherent diagrams in the sense of Vogt. This paper includes an explicit proof that the homotopy coherent nerve of a locally Kan simplicially enriched category is a quasicategory. As well as the harder result on when outer horns in this quasicategory can be filled.
Vogt’s original version of the theorem is in
Two other papers are relevant to this:
Jean-Marc Cordier, Tim Porter, Maps between homotopy coherent diagrams, Top. and its Appl. 28, (1988), 255 – 275.
Jean-Marc Cordier, Tim Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure. Applied Alg. 67, (1990), 111 – 124.
An elementary discussion of the concept of homotopy coherence forms Chapter V of
For the role played by the simplicial nerve in the context of relating quasi-categories to simplicially enriched categories as models for $(\infty,1)$-categories see
This emphasises the adjunction corresponding to the homotopy coherent (“simplicial”) nerve construction.
A review of this latter aspect is also in
Vivek Dhand, The simplicial nerve of a simplicial category (pdf)
Mitya Boyarchenko, Notes and exercise on $\infty$-categories (pdf)
Vladimir Hinich, Simplicial nerve in Deformation theory (arXiv:0704.2503)
Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal sets and simplicial operads, arxiv/1109.1004 (a Quillen equivalence for Segal vs. simplicial operads using coherent nerve)
Emily Riehl, On the structure of simplicial categories associated to quasi-categories, Math. Proc. Camb. Phil. Soc. 150 (2011), 489 - 504.
For more references see relation between quasi-categories and simplicial categories.
Two query-discussions on terminology and concrete description of the coherent/”simplicial” nerve are archived at nForum here.