nLab homotopy coherent diagram

Idea

A homotopy coherent diagram is a diagram of objects in a homotopical category, where commutativity is replaced by explicit homotopies, those homotopies are to then be coherently linked by higher homotopies … and so on.

It is a model for an (∞,1)-functor.

The idea perhaps intuitively makes sense but the management of the interactions between the various levels of homotopy requires care. The ideas were handled in various ways , but we will concentrate on approaches linked to the initial work of Michael Boardman and Rainer Vogt and then developed further by Jean-Marc Cordier and Tim Porter. There was another more-or-less equivalent approach by Dwyer and Kan, but this did not make the relationship with homotopy coherence quite so explicit.

We will often use h.c. as an abbreviation for ‘’homotopy coherent’’.

Definition

In components

The original definition of Vogt, 1973 is essentially the following.

Suppose now that we have the h.c. diagram $F : S(\mathbb{A}) \to \mathcal{B}$. This is specified by assignments:

• to each object $a$ of $\mathbb{A}$, it assigns an object $F(a)$ of $\mathcal{B}$;

• for each string of composable morphisms in $\mathbb{A}$,

$$\sigma = (f_0, \ldots, f_n)$$

starting at $a$ and ending at $b$, a simplicial map

$$F(\sigma) : S(\mathbb{A})(0,n+1) \to \mathcal{B}(F(a), F(b)),$$

that is, a higher homotopy

$$F(\sigma) : \Delta[1]^n \to \mathcal{B}(F(a), F(b)),$$

such that

(i) if $f_0 = id$, $F(\sigma) = F(\partial_0\sigma)(proj \times \Delta[1]^{n-1})$

(ii) if $f_i = id$, $0\lt i \lt n$

$$F(\sigma) = F(\partial_i\sigma(.(I^i \times m \times I^{n-i}),$$

where $m : I^2 \to I$ is the multiplicative structure on $I = \Delta[1]$ by the ‘max’ function on $\{0,1\}$;

(iii) if $f_n = id$, $F(\sigma) = F(\partial_n \sigma)(I^{n-1} \times proj)$;

(iv)$_{i}$ $F(\sigma)|(I^{i-1}\times \{0\} \times I^{n-i}) = F(\partial_i\sigma), 1 \leq i \leq n-1$;

(v)$_{i}$ $F(\sigma)|( I^{i-1}\times \{1\} \times I^{n-i}) = F(\sigma^\prime_i) . F(\sigma_i)$, where $\sigma_i = (f_0, \ldots, f_{i-1})$ and $\sigma^\prime = (f_i, \ldots, f_n)$. We have used $\partial_i$ for the face operators in the nerve of $\mathbb{A}$.

This original form can be very useful for checking (bare hands!) within an application that a diagram is h.c., although the $SSet$-functor approach is for many uses more compact and maniable and allows functorial constructions more easily. The link with the bar construction and comonadic resolution approaches give suggestive links to interpretation of cohomology classes.

As algebras over an operad.

For $\mathcal{E}$ a symmetric monoidal category, and $C$ a small $\mathcal{E}$-enriched category, there is an operad $Diag_C$ whose algebras over an operad are $\mathcal{E}$-enriched functors

$$F : C \to \mathcal{E}$$

hence $C$-diagrams in $\mathcal{E}$.

If $\mathcal{E}$ is also a monoidal model category with an interval object $H$ in a sufficiently nice way, then there exists the Boardman-Vogt resolution

$$HoCoDiag_C := W(H, Diag_C) \,.$$

The algebras over this operad are then precisely homotopy coherent diagrams over $C$ in $\mathcal{E}$. For $\mathcal{E} =$ Top regarded with the standard model structure on topological spaces and $H = [0,1]$ the standard interval, this reproduces the ordinary notion of homotopy coherent diagrams (BergerMoerdijk)

Properties

Equivalence

(i) If $X : \mathbf{A}\to$ Top is a commutative diagram and we replace some of the $X(a)$ by homotopy equivalent $Y(a)$ with specified homotopy equivalence data:

$$f(a) : X(a) \to Y(a), \quad g(a) : Y(a) \to X(a)$$

$$H(a) : g(a)f(a) \simeq Id, \quad K(a) : f(a)g(a) \simeq Id,$$

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

$$f : X\to Y, \quad g : Y \to X, etc.,$$

making $X$ and $Y$ homotopy equivalent as h.c. diagrams.

(This applied to a $G$-space, $X$, shows that if we replace $X$ by a homotopy equivalent $Y$, then $Y$ will be a h. c. version of a $G$-space, i.e. a h. c. diagram of shape $BG$, the corresponding one object groupoid to $G$.)

Cordier’s Homotopy coherent nerve

Theorem

Cordier (1980)

For each a small category $\mathbb{A}$, the sSet-enriched category ${S(\mathbb{A})}$ defined in homotopy coherent nerve#the_cosimplicial_category is such that a h.c. diagram of shape ${\mathbb{A}}$ in Top is given precisely by an sSet-enriched functor

$$F : {S(\mathbb{A})} \to Top$$

This suggested the extension of h.c. diagrams to other contexts such as a general locally Kan $SSet$-category, $\mathcal{B}$ and further suggests the definition of homotopy coherent diagram in a $\mathcal{S}$-category and thus a homotopy coherent nerve of an $SSet$-category. This was first done by Cordier and Porter in 1986, (see references). If the $SSet$-category is “locally Kan”, this homotopy coherent nerve is a quasicategory.

To understand simplical h.c. diagrams and thus the h.c. simplicial nerve $N(\mathcal{B})$, we unpack the definition of homotopy coherence, for convenience, repeating some points made in homotopy coherent nerve.

The first thing to note is that for any $n$ and $0\leq i\lt j\leq n$, $S[n](i,j) \cong \Delta[1]^{j-i-1}$, the $(j-i-1)$-cube given by the product of $j-i-1$ copies of $\Delta[1]$. Thus we can reduce the higher homotopy data to being just that, maps from higher dimensional cubes.

Next some notation:

Given simplicial maps

$$f_1: K_1 \to \mathcal{B}(x,y),$$

$$f_2: K_2 \to \mathcal{B}(y,z),$$

we will denote the composite

$$K_1 \times K_2 \to \mathcal{B}(x,y)\times \mathcal{B}(y,z) \stackrel{c}{\to} \mathcal{B}(x,z)$$

just by $f_2.f_1$ or $f_2f_1$. (We will seen this in the h.c. diagram below for $\mathbb{A} = [3]$. $X(123)X(01)$ is actually $X(123)(I \times X(01) )$, whilst $X(23)X(012)$ is exactly what it states.)

Rectification

Every homotopy coherent diagram is weakly equivalent to a strict diagram, a phenomenon known as rectification.

Vogt’s theorem

Theorem

(Vogt)

If $\mathbf{A}$ is a small category, there is a category $\mathbf{Coh(A,Top)}$ of h.c. diagrams and homotopy classes of h. c. maps between them. Moreover there is an equivalence of categories

$$\mathbf{Coh(A,Top)} \stackrel{\simeq}{\to} \mathbf{Ho(Top^A)}.$$

Berger-Moerdijk theorem

If we think of hc diagrams as algebras over an operad, then this rectification is a special case of the general rectification theorem for such algebras. See model structure on algebras over an operad for details.

Examples

h.c. diagrams in a category with cylinder functor, denoted $-\times I$

1. A diagram indexed by the small category, $[2]$.

is h.c. if there is specified a homotopy

$$X(012) : X(0)\times I \to X(2),$$

$$X(012) : X(02) \simeq X(12)X(01).$$

1. For a diagram indexed by $[3]$: Draw a 3-simplex, marking the vertices $X(0), \ldots, X(3)$, the edges $X(ij)$, etc., the faces $X(ijk)$, etc. The homotopies $X(ijk)$ fit together to make the sides of a square

$$\begin{matrix} X(1 3)X(0 1)&\xrightarrow{X(1 2 3)X(0 1)}&X(2 3)X(1 2)X(0 1)\\ \mathllap{X(0 1 3)}\left\uparrow\space{30}{20}{0}\right.& &\left.\space{30}{20}{0}\right\uparrow\mathrlap{X(2 3) X(0 1 2)}\\ X(0 3)&\xrightarrow[\qquad X(0 2 3)\qquad]{\quad}&X(2 3)X(0 2) \end{matrix}$$

and the diagram is made h.c. by specifying a second level homotopy

$$X(0123) : X(0)\times I^2\to X(3)$$

filling this square, in the sense that restricting to each side of the square in the double homotopy gives the correspondingly labelled homotopy from the diagram.

These can be continued for larger $[n]$, and the results glued together to make larger h.c. diagrams. Of course, this is not how it is actually done, but may provide some help in understanding the basic idea.

References and Literature

For Vogt‘s theorem, the original reference is

• Rainer Vogt, Homotopy limits and colimits, Math. Z., 134 (1973) 11-52 [doi:10.1007/BF01219090, eudml:171965]

A generalisation of his theorem using simplicially enriched categories and the homotopy coherent nerve of such a thing, is to be found in

• J.-M. Cordier and T. Porter, Vogt’s Theorem on Categories of Homotopy Coherent Diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986) pp. 65-90.

A neat application to changing objects in diagrams within a homotopy type can be found in

• J.-M. Cordier and T. Porter, Maps between homotopy coherent diagrams, Top. and its Appls., 28, (1988), 255 – 275.

See also

• William Dwyer, Dan Kan, Jeff Smith, Homotopy commutative diagrams and their realizations, Journal of Pure and Applied Algebra 57 (1989) 5-24 (doi:10.1016/0022-4049(89)90023-6)

A summary of homotopy coherence can be found in Chapter 5 of

• Klaus Heiner Kamps, Tim Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific 1997 (doi:10.1142/2215, GoogleBooks)

and in chapter 10 of

• Tim Porter, Crossed Menagerie

The operad-theoretic description of homotopy-coherent diagrams is in

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)

See model structure on algebras over an operad for more on this.