Homotopy Limit Intuition

Question

I am currently reading through Emily Riehl's Categorical Homotopy Theory book, and have gotten a bit stuck on the section on calculating homotopy limits. The colimit case is fine, but for some reason I am struggling with the intuition for homotopy limits. I am trying to work through the exercise where I compute the homotopy pullback of $p: Y \to X, q: Z \to X$, using the cobar construction. Could someone give me a little help/intuition for how to tackle these problems?

Answer 1

Well, asking for "the intuition" is a bit different from asking how to compute this using the cobar construction. Most small examples are not clarified by that generality-this exercise is more intended (IMO) to improve your intuition for the cobar construction by leaning on pre-existing intuition for this homotopy limit than the other way around.

So what is the pre-existing intuition you may be missing? The homotopy limit of your diagram is the homotopy pullback of $Y$ with $Z$ over $X$. As the strict pullback represents pairs of maps into $Y$ and $Z$ which become equal in $X$, the homotopy pullback represents pairs of maps into $Y$ and $Z$, together with the choice of a homotopy between them in $X$.

For instance if $X$ is a based space and $Y$ and $Z$ are the point, then the homotopy pullback effectively represents choices of homotopies from the base point of $X$ to itself. In this case the pullback is traditionally denoted $\Omega X$, the space of based loops in $X$. If $X$ lives in any category with a notion of homotopy and $Y$ and $Z$ are both $X$, which $p$ and $q$ the identities, then the homotopy pullback represents pairs of maps into $X$ together with a homotopy between them. In this case it is traditionally denoted $X^I$, the space of paths in $X$. For a third example, if $q$ is any map of based spaces and again $Y$ is the point, then the homotopy pullback effectively represents maps into $Z$ with a chosen nullhomotopy in $X$. This object is traditionally called the homotopy fiber of $q$.

So in complete generality, if $W$ is the homotopy pullback of $p$ and $q$, you can construct $W$ as a subspace of $Y\times Z\times X^I$, which in the case of a category in which points of objects make sense is given by the space of triples $\{(y,z,\gamma)\}$, where $\gamma$ is a path in $X$ from $p(y)$ to $q(z)$. Hopefully that clarifies the concept enough to let you realize it as an example of a cobar construction.