homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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.
We will often use h.c. as an abbreviation for ”homotopy coherent”.
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 $\mathbb{A}$,
starting at $a$ and ending at $b$, a simplicial map
that is, a higher homotopy
such that
(i) if $f_0 = id$, $F(\sigma) = F(\partial_0\sigma)(proj \times \Delta[1]^{n-1})$
(ii) if $f_i = id$, $0\lt i \lt n$
where $m : I^2 \to I$ is the multiplicative structure on $I = \Delta[1]$ by the ‘max’ function on $\{0,1\}$;
(iii) if $f_n = id$, $F(\sigma) = F(\partial_n \sigma)$;
(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.
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
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
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)
(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:
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
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 (1980)
Given ${\mathbb{A}}$, a small category, the sSet-enriched category ${S(\mathbb{A})}$ called the homotopy coherent nerve is such that a h.c. diagram of shape ${\mathbb{A}}$ in Top is given precisely by an sSet-enriched functor
This suggested the extension of h.c. diagrams to other contexts such as a general locally Kan $SSet$-category, $\mathcal{B}$ and suggests the definition of homotopy coherent diagram in a $\mathcal{S}$-category and thus a homotopy coherent nerve of an $SSet$-category.
To understand simplical h.c. diagrams and thus the h.c. simplicial nerve $N(\mathcal{B})$, we unpack the definition of homotopy coherence, for conveneince, 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
we will denote the composite
just by $f_2.f_1$ or $f_2f_1$. (We already have seen this in the h.c. diagram above 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.)
Every homotopy coherent diagram is weakly equivalent to a strict diagram, a phenomenon known as rectification.
(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
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.
h.c. diagrams in a category with cylinder functor, denoted $-\times I$
is h.c. if there is specified a homotopy
and the diagram is made h.c. by specifying a second level homotopy
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 done, but it provides some understanding of the basic idea.
For Vogt’s theorem, the original reference is
A generalisation of his theorem using simplicially enriched categories and the homotopy coherent nerve of such a thing, is to be found in
A neat application to changing objects in diagrams within a homotopy type can be found in
A summary of homotopy coherence can be found in Chapter 5 of
and in chapter 10 of
The operad-theoretic description of homotopy-coherent diagrams is in
See model structure on algebras over an operad for more on this.