In looking at phenomena in a category such as that of spaces and continuous maps, which has a notion of homotopy, one often needs to replace a space in some diagram by another, perhaps more ‘manageable’ one, but within the same homotopy type, or to replace a structural map in a diagram by a homotopic one that has nicer properties. This, however, may destroy the commutativity of the diagram making it a ‘homotopy commutative diagram’, however, if one then needs to take a limit (or colimit) of such a diagram, one cannot with impunity! The limit may no longer exist since the diagram may not commute, and the ‘maps’ in it are now homotopy classes, but even if one can find a commutative diagram representing it the (homotopy type of the) limit may depend on what representing diagram one takes.
There is a very simple example of this latter phenomenon. Suppose $X$ is an arcwise connected space, and $i: A\to X$ the inclusion of a proper subspace, and work out the homotopy type of the fibre over a point $x\in X$. If we denote the terminal singleton space by $\top$, there is a unique map $x:\top \to X$, sending the unique point of $\top$ to $x$. Changing this map within its homotopy class corresponds to changing $x$ along a path in $X$, but the fibre is a limit of a diagram, namely the pullback of $i$ along $x$, and its homotopy type will change as the path joining some $x$ to another wanders in and out of $A$. The homotopy type of the fibre is dependent on the actual point taken not just on the homotopy class of the representing map in the pullback diagram.
Another example is with spaces of loops at a point $x$ in a space $X$. the elements in the loop space are maps from a circle, $S^1$, to $X$ that send $1\in S^1$. This has the structure of a group ‘up to homotopy’, but better ‘up to coherent homotopy’.
We will try to give some insight into the group of ideas associated with this. For more details, look up homotopy coherence, homotopy coherent diagram, homotopy coherent nerve, and other similar terms on the nlab.
Imagine a diagram of spaces in the form of a tetrahedron. There are spaces, $X(o)$, \ldots, $X(3)$, one for each vertex; for each $i\lt j$, a map $X(i,j): X(i)\to X(j)$.
and the diagram is made h.c. by specifying a second order homotopy
(We have written $X(1,2,3)X(0,1)$ as a short form of the composite of $X(0,1)\times I: X(0)\times I \to X(1)$ with $X(1,2,3): X(1)\times I \to X(3)$.)
The problem of giving an understandable description of homotopy coherence is technically hard, due to the need not only to handle morphisms and homotopies between them, but then homotopies between homotopies, and homotopies between those, and so on. It is a problem that is combinatorial in its nature, but needs a mix of combinatorial and topological insights in its solution. It is a key problem for the understanding of several important areas of mathematics and mathematical physics.
Boardman and Vogt gave a topological category based theory, simplified by Vogt (1974). Work in shape theory by Tim Porter (1974) suggested the need for a simpler, more categorical and combinatorial version of their theory. The point was the classical problem of refinement in the construction of the Čech nerve of an open covering in the definition of Čech homology, or, more recently, in descriptions of Borsuk’s shape theory or for the . There was a sense in which the inverse system of étale homotopy theory of Artin and Mazur, (1969). There had to be a choice of refinement map between related open covers of a space, but the resulting diagram of simplicial complexes or chain complexes, was only homotopy commutative, as these choices influenced the various compositions. The result could be shown to be representable, in the shape theory situation, by an actual commutative diagram, the Vietoris complex, but at the cost of using enormous complexes and losing the geometric insight. Boardman and Vogt’s theory indicated that this meant that the Čech complex must be homotopy coherent, but again at a cost of going from simplicial sets to spaces and back again, thus loosing both precision and, once again, geometric insight. What was needed was a simplicial version of Vogt’s theory.
Jean-Marc Cordier (1980) came up with a good simplicial categorical theory and clarified its relationship with Vogt’s theory via a homotopy coherent nerve. The collaboration with Porter on the detailed study of that nerve, and exploration of a homotopy coherent version of category theory, and on its applications to problems in algebraic topology started at this time.
In about 1983, the importance of this project was considerable enhanced when, in correspondence with Grothendieck, he revealed that in letters to Larry Breen in the mid 1970s, he had loosely conjectured the existence of a higher order version of the theory of stacks, which would have application in higher order non-Abelian cohomology, would also generalise the ‘Poincaré-Galois theory’ giving the links between the fundamental group and covering spaces (SGA1) and hence give a higher dimensional version of Galois theory. One of the key parts of this conjectural theory would be a form of homotopy coherence or weak infinity category theory whose form would be very much as Cordier and Porter had started developing. (Another part of the conjectured theory would involve finding algebraic or categorical models for homotopy n-types.)
The main results of this Cordier-Porter theory of homotopy coherence were:
The homotopy coherent nerve of a simplicial category, $C$, is a weak Kan complex, i.e., in more modern terms, a quasicategory, provided that for each pair, $(x,y)$, of objects in $C$, the simplicial set, $C(x,y)$ is a Kan complex.
In such a case provided that $C$ is complete and cocomplete, an important generalisation of Vogt’s theorem holds. This identifies the homotopy category of any diagram category, $C^I$ , in terms of a category consisting of homotopy coherent diagrams of shape $I$ in $C$. The proof used an induction up the skeleton of the homotopy coherent nerve, so a geometricly inspired combinatorial method, together with new categorical methods of rectification of homotopy coherent diagrams which have since become standard. It also proved for the first time various results on quasi-categories which have since been used by Joyal (2002) and by Lurie (Annals Math studies, 2009) in acclaimed work on higher dimensional categories, quasi-categories, and analogues of topos theory in that context.
In the 1980s, Cordier and Porter published several papers linking their methods with those available using other more case specific approaches and then in 1997 they published a 50 page paper in which much of the basic theory of categories was extended to the homotopy coherent context. This has since become a standard reference for work on homotopy coherent ends and coends, coherent Kan extensions etc. (The basic work was done in 1984-6, but publication had been delayed.)
One important proof involving the ordinal sum construction in that 1997 paper was expanded and explored further in his 1993 PhD thesis by Phil Ehlers, who was Porter’s PhD student. This was pushed further forward by Porter, and the result published as a joint paper in Advances in Maths (2007). This aspect has further been used by Verity and others to explore new structures in the hard area of models of infinity categories. Verity also makes extensive use of the homotopy coherent nerve and its properties in his work extending that of Street on weak infinity categories. This theory has become a standard tool for workers in higher category theory and its applications in mathematical physics and algebraic geometry.