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My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:

  • Cordier and Porter proved a rectification theorem for diagrams in Kan-enriched categories $\mathcal{M}$ satisfying certain additional conditions: for every small category $\mathbb{D}$, if $\mathbf{Coh}(\mathbb{D}, \mathcal{M})$ is the homotopy category of homotopy-coherent diagrams $\mathbb{D} \to \mathcal{M}$ and $\operatorname{Ho} {[\mathbb{D}, \mathcal{M}]}$ is the homotopy category of strict diagrams $\mathbb{D} \to \mathcal{M}$, then the obvious functor $\operatorname{Ho} {[\mathbb{D}, \mathcal{M}]} \to \mathbf{Coh}(\mathbb{D}, \mathcal{M})$ is an equivalence of categories. This holds in particular for $\mathcal{M} = \mathbf{Top}$ (originally proved by Vogt) and $\mathcal{M} = \mathbf{Kan}$.
  • It is well-known that every monoidal category is monoidally equivalent to a strict monoidal category: this is one of the consequences of Mac Lane's coherence theorem.
  • A. J. Power proved a rectification theorem for algebras for 2-monads: given a 2-monad $\mathsf{T}$ on the 2-category $\mathfrak{Cat}$ that preserves bijective-on-objects functors, every pseudo-$\mathsf{T}$-algebra is equivalent to a strict $\mathsf{T}$-algebra in the 2-category of pseudo-$\mathsf{T}$-algebras and strong homomorphisms.

What is the state of the art in the rectification of homotopy-coherent structures, in the general sense suggested above?

I am especially interested to hear about results concerning the rectification of homotopy-coherent diagrams in (not necessarily simplicial) model categories. This appears to be a crucial step toward showing that homotopy limits and colimits in the sense of derived functors agrees with homotopy limits and colimits in the Joyal–Lurie sense. (See e.g. §4.2.4 in [Higher topos theory].)

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1 Answer 1

One general rectification result is that for algebras over an operads, as in

  • Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)

which is briefly summarized on the nLab at model structure on algebras over an operad -- Properties -- Rectification of algebras.

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I thought there might be an operadic result of this kind, thanks. Does the operad for (homotopy) commutative diagrams satisfy the technical conditions? –  Zhen Lin Sep 4 '13 at 7:49
Yes, this is the example in section 5 of the above article by Berger-Moerdijk. –  Urs Schreiber Sep 4 '13 at 9:47
On the nLab this is here (very briefly): ncatlab.org/nlab/show/… –  Urs Schreiber Sep 4 '13 at 9:48

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