ScalingStacks

We note that several other abstract models of homotopy theory have been proposed. One has been proposed by W. Dwyer and D. Kan, as was noted above. Since the complete Segal spaces described in our work are themselves objects in a certain closed model category, our construction gives another model for a homotopy theory of homotopy theory. We believe that our model is “equivalent” to that of Dwyer and Kan, via a suitable notion of equivalence; in particular, there should be constructions which take complete Segal spaces to simplicially enriched categories and vice versa, and these constructions should be inverses to each other (modulo appropriate notions of equivalence.) We hope to give a proof of this in the future.

Another model has been proposed by A. Heller [Hel88]. He suggests that a homotopy theory be modeled by a certain type of contravariant $2$-functor from the category of small categories to the category of large categories. For example, from a closed model category ${\operatorname{\mathbf{M}}}$ there is a construction which assigns to each small category $C$ the homotopy category $\ho({\operatorname{\mathbf{M}}}^{C})$ of the category of $C$-diagrams in ${\operatorname{\mathbf{M}}}$, and which associates to each functor $C\rightarrow D$ restriction functors $\ho({\operatorname{\mathbf{M}}}^{D})\rightarrow\ho({\operatorname{\mathbf{M}}}^{C})$ which themselves admit both left and right adjoints, arising from “homotopy Kan extensions”. Because Heller’s models require the existence of such homotopy Kan extensions, they seem to be less general than the models considered in this paper, and we do not know the proper relationship between his theory and the others.