Here's a definition for homotopy limits that isn't quite right, but seems salvageable. Does anyone know how to fix it?
Suppose the category $C$ is some reasonable setting for homotopy theory, say it's enriched over some kind of category of spaces (e.g. chain complexes, simplicial sets, ...).
Def: Let F: Dop $\to$ C be a diagram (functor). An object X together with a map η from X to the diagram is a LIMIT for the diagram iff the induced natural transformation of functors HomC(-, X) $\to \lim$ HomC(-,F) is an isomorphism. A pair (X,η) is a homotopy limit for the diagram F iff the induced transformation of functors HomC(-,X) $\to \lim$ HomC(-, F) is a weak equivalence.
This definition doesn't quite cut it since, in most of the motivating examples I know, though the homotopy limit object X does come equipped with a morphism to each object in the diagram, these do not commute with the morphisms in the diagram---they only commute up to homotopy. So a homotopy limit won't even come with a map to the diagram, so it doesn't come with an induced natural transformation. How then can I characterize the object X by a similar universal property as the (strict) limit?