Holmstrom Homotopy colimit

Much work by Thomason. Notes from Weibel’s obituary: Thomason constructed, for a diagram $J \to C$ of small categories, a canonical cofibered category such that it’s geometric realization is the homotopy colimit of the geometric realization of the diagram. Remark: An infinite loop space machine sends homotopy colimits in SymMonCat to the corresponding hocolims of spectra.

arXiv:1104.0646 Realizable homotopy colimits from arXiv Front: math.AG by Beatriz Rodriguez-Gonzalez In this paper we study realizable homotopy colimits, which we define as homotopically well-behaved cofibrant approximations of colimits. Homotopical well-behavior includes cofinality, Fubini and homotopy invariance, as well as existence of pointwise homotopy left Kan extensions. We characterize such realizable homotopy colimits on (D,E) as those hocolim obtained as the composition of the simplicial replacement with a simple functor endowing (D, E) with a simplicial descent structure (arXiv:0808.3684). As an example, we deduce that (finite) homotopy limits exist for mixed Hodge complexes, and are realized through Deligne’s cosimplicial construction.

http://mathoverflow.net/questions/26932/simple-examples-of-homotopy-colimits

http://mathoverflow.net/questions/33556/do-homotopy-colimits-always-commute-with-homotopy-colimits

http://mathoverflow.net/questions/33561/sequential-colim-vs-sequential-hocolim

http://mathoverflow.net/questions/37992/the-homotopy-cofiber-of-the-smash-product-of-two-maps-of-spectra

http://mathoverflow.net/questions/82813/does-the-right-adjoint-of-a-quillen-equivalence-preserve-homotopy-colimits

http://mathoverflow.net/questions/55271/when-does-a-cosimplicial-object-compute-homotopy-colimits

http://mathoverflow.net/questions/38047/what-is-the-universal-problem-that-motivates-the-definition-of-homotopy-limits

arXiv:1109.0265 Homotopy Colimits of Algebras Over Cat-Operads and Iterated Loop Spaces from arXiv Front: math.AT by Zbigniew Fiedorowicz, Manfred Stelzer, Rainer M. Vogt We extend Thomason’s homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the category of $n$-fold loop spaces and loop maps after localization with respect to some other class of weak equivalences. We recover Thomason’s corresponding result about infinite loop spaces and obtain related results about braided monoidal categories and 2-fold loop spaces.

Jardine-Goerss talks early in chapter IV about the homotopy colimit of a functor into simplicial sets; this hocolim is a bisimplicial set. For bisimplicial abelian groups this hocolim appears naturally “in any homology ss arising from the hocolim of a diagram of simplicial sets”.

nLab page on Homotopy colimit