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?
 A: Reid's answer is quite right, but long before "quasicategories" became fashionable, algebraic topologists were doing exactly the same thing using the "simplicial bar construction" and plain old topological or simplicial enriched categories.
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.
A: You can fix it by making η a homotopy coherent diagram (a map to each object of D, a homotopy for each arrow of D, a homotopy-between-homotopies for each commuting triangle in D, ...) and also replacing lim HomC(–, F) with a suitable space of homotopy coherent diagrams.  This is essentially what you do in the quasicategory-theoretic definition of (homotopy) limit; see section 1.2.13 of Higher Topos Theory for a brief description.
