Homotopy limit

Remarks

The homotopy limit sends objectwise weak equivalences of diagrams to weak equivalences, and it comes with a natural map $\lim D \to \operatorname{holim} D$.

Example: for a diagram $X \xrightarrow{f} Z \xleftarrow{g} Y$ of topological spaces, the homotopy limit is the homotopy pullback $$X \times_Z Z^{[0,1]} \times_Z Y = \set{(x, \omega, y) : \omega \colon [0, 1] \to Z,\ \omega(0) = f(x),\ \omega(1) = g(y)}.$$ When $f$ or $g$ is a Serre fibration, the map from the ordinary pullback to the homotopy pullback is a weak equivalence.

Non-example: the ordinary limit is not homotopy invariant. There is a map from the diagram $\set{0} \to \R \leftarrow \set{1}$ to the diagram $\set{0} \to \R \leftarrow \set{0}$ that is the identity on $\set{0}$, sends $1$ to $0$, and is $t \mapsto t^2 - t$ on $\R$. It commutes with the maps of the diagrams and is a weak equivalence at each object. The pullback of the first is empty and that of the second is a point, while both homotopy pullbacks are contractible path spaces.

For a diagram $D$ of Kan complexes there is an explicit model, the Bousfield–Kan formula $\operatorname{holim} D = \int_{i \in I} \mathrm{Map}(N(I/i), D(i))$, where $N(I/i)$ is the nerve of the over category. Homotopy colimits are defined dually, as the left derived functor of the colimit, using the projective model structure.