# References for homotopy colimit

(1) What are some good references for homotopy colimits?

(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will think of as a category and also as a directed graph (objects = vertices, morphisms = edges). Assume we have a functor $F$ from the graph into (say) chain complexes. We will construct a big chain complex (the homotopy colimit) in stages.

Stage 0: direct sum over all vertices $v$ of $F(v)$

Stage 1: direct sum over all edges $e$ of the mapping cylinder of $F(e)$, with the ends of the mapping cylinder identified with the appropriate parts of stage 0.

Stage 2: direct sum over all pairs of composable edges $(e_1, e_2)$ of a higher order mapping cylinder, with appropriate identifications to parts of stage 1. This implements a relation between the three stage 1 mapping cylinders corresponding to $e_1, e_2$ and $e_1*e_2$.

Stage 3: direct sum over all triples of composable edges $(e_1, e_2, e_3) \dots$

I'm way late on this one, but for the record I'll point out that a nice answer to question 2 can be found in Hatcher's Algebraic Topology book, Section 4.G.

Dan Dugger wrote the following intended for grad students (just a draft - not on the arxiv yet): http://www.uoregon.edu/~ddugger/hocolim.pdf

• The link is dead, but the paper is available from Wayback Machine – Wojowu Jul 30 '19 at 9:14

(1) I happen to like this paper, but of course I'm biased. (I hope self-citation isn't forbidden here...) However, I started writing that paper mainly because I couldn't find an existing reference/introduction that I liked. So if someone has another reference to suggest I would love to hear about it.

(2) This sounds like the simplicial bar construction, which is the one I used in my paper above. I think I included some other references in the bibliography.

Chris Douglas has a nice short discussion of homotopy limits in his text “Sheaves in homotopy theory” (Chapter 5 of Topological modular forms).

A good (if kind of old) reference is Vogt's "Homotopy Limits and Colimits". I can't find a free reference for it, but if you can't access it, I could email a pdf (if that's allowed here).

Also, in the IMA's video library there's a video of Gunnar Carlsson giving a talk on homotopy limits & colimits. http://www.ima.umn.edu/videos/?id=870

• "I could email a pdf (if that's allowed here)." Yes, that is fine. – David Zureick-Brown Oct 14 '09 at 21:50

Some useful information is in

Chacholski, Scherer, Homotopy theory of diagrams, arxiv math/0110316

This and further references with useful material are listed in the reference section at nLab: homotopy limit.

(By immediate dualization this applies to homotopy colimits, of course).

I found the following exposition by Emily Riehl very useful:

http://www.math.jhu.edu/~eriehl/hocolimits.pdf

I learned today that Pascal Lambrechts also wrote a short primer on homotopy limits and colimits, filled with good exercises. It is online here (for now).

I have learned a great deal from Emily Riehl's book, Categorical Homotopy Theory. At the time of this answer a (possibly outdated) pdf is hosted on Emily's website, here. Chapter 6 is full of useful information. I also enjoyed the examples in section 8 of her paper The Theory and Practice of Reedy Categories with Dominic Verity.

• Regarding Riehl's book: "CUP has graciously allowed me to host a free PDF copy in perpetuity, which will eventually be updated to the final copyedited version." My guess is that what's there currently is the final draft. – David Roberts Jun 16 '15 at 7:45
• "Theory and Practice of Reedy categories" is such a good paper. I am not an algebraic topologist, but it really gave me a feel for how homotopy limits / colimits work – Daniel Barter Jun 16 '15 at 12:01

My answer may be a bit late, but E. E. Floyd and W. J. Floyd: Actions of Classical Small Categories has helped me understand some of these concepts.