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 . This is specified by assignments:
where is the multiplicative structure on by the ‘max’ function on ;
(iii) if , ;
(v) , where and . We have used for the face operators in the nerve of .
This original form can be very useful for checking (bare hands!) within an application that a diagram is h.c., although the -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.
This suggested the extension of h.c. diagrams to other contexts such as a general locally Kan -category, and suggests the definition of homotopy coherent diagram in a -category and thus a homotopy coherent nerve of an -category.
To understand simplical h.c. diagrams and thus the h.c. simplicial nerve , 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 and , , the -cube given by the product of copies of . 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 or . (We already have seen this in the h.c. diagram above for . is actually , whilst is exactly what it states.)
Every homotopy coherent diagram is weakly equivalent to a strict diagram, a phenomenon known as rectification.
If is a small category, there is a category 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.