Some time ago I asked this question. I am trying again to get an understanding of the definition of the homotopy colimit of a diagram of topological spaces.
One of the answers at the above says
For example, homotopy colimits represent "homotopy coherent cones"
I am finding difficulty with the high level of generality of the definitions in both of these sources. My first question is whether there is a simpler definition of a homotopy coherent cone when it is over a finite diagram of CW-topological spaces - in particular where the maps are all cofibrant inclusions?
I am particularly interested in whether my following intuition is correct:
A homotopy coherent cone on a diagram is one such that we do not necessarily have commutativity, but we have commutativity up to homotopy, and then commutativity of those homotopies up to homotopy, and then commutativity of those up to homotopy, etc.
If this is not correct, is there a similar intuition available for homotopy coherent cones?