# Definition of homotopy limits

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?

For a fixed x, the limit $\lim \hom_C(x,F)$ is equivalent to the set of natural transformations from the constant functor $\Delta_1\colon D\to Set$ at $1$ to the functor $\hom_C(x,F(-))$. So to replace it by something "coherent" we need a notion of "homotopy coherent transformation." Now the set of natural transformations from a functor $G\colon X\to Top$ to a functor $H\colon Y\to Top$ can be defined as an "end," and computed as an equalizer of the two maps $\prod_x \hom(G x, H x) \rightrightarrows \prod_{x \to y} \hom(G x, H y)$. But these two maps are the first two coface maps of a cosimplicial object that continues with $\prod_{x\to y\to z} \hom(G x, H z)$ and so on, so we can define the space of "homotopy coherent transformations" to be its totalization (the dual of geometric realization).
Now a homotopy limit can be defined as a representing object for the space of homotopy coherent transformations from $\Delta_1$ to $\hom_C(x,F(-))$ (where now $C$ is topologically enriched, so that these functors take values in spaces). Moreover, if $C$ admits "totalizations" as a topologically enriched category (a sort of "weighted limit"---it suffices to have ordinary limits and "cotensors"), then the homotopy limit can be constructed by "internalizing" the above construction.