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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"

referring me to papers by Michael Shulman and Emily Riehl for the definitions.

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?

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Yes, that's exactly the right intuition. For certain simple diagrams, the legitimate colimit is homotopy equivalent to the homotopy colimit, so can be used in place of the more complicated construction. For instance, this is true for homotopy pushouts when the given maps are cofibrations. In general, diagrams require more than merely that every map be a cofibration to have this property. They must be cofibrant in the projective model structure, which can be complicated in general. However, it's easy to understand for simple categories like the arrow and the span, and it's a good idea to understand this construction in some simple cases to feel a better intuition for homotopy colimits. It's also helpful to study the global perspective, that a homotopy colimit functor is simply a derived colimit functor, to complement the heavier homotopy coherent cone concept.

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  • $\begingroup$ To be clear, for something to be a homotopy limit or colimit, must it itself be a homotopy coherent cone on the diagram in question? I believe that the answer to this is yes. Secondly, for example in the case of a homotopy pullback, does the homotopy pullback of a diagram come with a homotopy between the two sides of a square, or do we simply say that there exists a homotopy and any homotopies are homotopic? $\endgroup$ – Matt Aug 26 '19 at 1:26
  • $\begingroup$ Yes, it must be coherent, which implies that the homotopies come as part of the structure. For more complicated diagram shapes coherent cocones involve higher homotopies between homotopies, and all of these must be given. The special case when the ordinary colimit works is when all these homotopies are taken to be trivial, but without some (co)fibrancy conditions this may not leave enough wiggle room for the homotopical universal property. $\endgroup$ – Kevin Arlin Aug 26 '19 at 10:49
  • $\begingroup$ I see. Thank you. One last question; Could it be said then that at the ncatlab page on homotopy pullbacks: ncatlab.org/nlab/show/homotopy+pullback the definition may not be stated accurately? They say there that the square must commute up to homotopy, but do not say that a homotopy is given. Is this strictly correct? $\endgroup$ – Matt Aug 26 '19 at 12:33
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    $\begingroup$ @Matt I think the intended interpretation is that a homotopies commutative square be a square with given homotopy, since otherwise asking for the space of homotopy commutative squares doesn't make sense. But I agree this is a potentially misleading exposition. $\endgroup$ – Kevin Arlin Aug 26 '19 at 16:30

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