Combinatorial Model Categories Have Presentations

By Daniel Dugger. 2000

225 citations recorded by the source corpus.

Abstract We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where “diagram category” means diagrams of simplicial sets). This says that every combinatorial model category can be built from a category of “generators” and a set of “relations.”

Subfields: Artificial Intelligence, Computer Networks and Communications, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4dc21b9aa7f6a4b33e3f3e8788ae745737c621ab

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Citations in the corpus

Spaces of smooth embeddings and configuration categories (2015, background): “pre-composition with λ:E0(x) →E1(x) does not aﬀect the forgetful functor x→y←z →z it follows that the inclusion hocolim x→y←z in E1(x) Y(z) −−−→ hocolim x→y←z in E0(x) Y(z) is also a weak equivalence [12]. The assignment x→E1(x) is not contravariantly functorial on all of simp(Z), but it is well beh”
Universal Homotopy Theories (2000, background): “We announce in (6.3) the result that every combinatorial model category is equivalent to a localization of some UC; the proof is too involved to include here, but is instead given in the companion paper [ D3 ].”
HZ -algebra spectra are differential graded algebras (2002, background): “Criteria (c0) and (c3) follow from [Be00, 1.15, 1.18, 1.19] since the functors evn: Sp (ch+)! ch+ are accessible and ch+ is locally presentable by Proposition 3.7; see also [ Du01b , 7.3] and [AR94, 2.23, 2.37].”
Topological equivalences for differential graded algebras (2006, background): “A model category is called combinatorial if it is cofibrantly-generated and the underlying category is locally presentable. See [ D1 ] for more information.”
K-THEORY AND DERIVED EQUIVALENCES (2002, background): “Finally, in Section 7 we develop the many-generators version of tilting theory, and prove Theorem D .”
Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, background): “Of course, these model structures have a long history going back to Bousfield-Kan and others, more recently considered by Hirschhorn [116], Smith (see Beke [27], Dugger [84]), Blander [41], Barwick [16].”
The homotopy theory of dg-categories and derived Morita theory (2004, background): “However, it is well known that there exist triangulated functors D(A) −→ D(B) that do not arise from a A op ⊗ B-dg-module (see e.g. [ Du -Sh, 2.5, 6.8]).”
Pro-categories in homotopy theory (2015, methodology): “simplicial combinatorial model categories and presentable ∞-categories (see [Lu09, Proposition A.3.7.6]). The theory can be extended to general combinatorial model categories using the work of Dugger [Du01] (see [Lu14, Propositions 1.3.4.22, 1.3.4.23 and 1.3.4.24]). However, it seems that a complete”
Homotopical algebraic geometry. I. Topos theory. (2002, methodology): “We like to consider this result as an extension of D. Dugger characterization of combinatorial model categories ([ Du2 ]), and as a model category analog of J. Lurie’s theorem characterizing ∞-topoi (see [Lu, Thm.”
Generalized Brown representability in homotopy categories (2005, unknown): “Probably, there is an alternative way of using [12] to get that each object of Ho(K) is a canonical homotopy λ-filtered colimit of objects from Ho(Kλ).”
Higher Topos Theory (2006, background): “We now complete the proof by invoking the following result, proven in [19]: for every combinatorial model category A, there exists a small category D, a set S of morphisms of Set op ∆ , and a Quillen equivalence of A with S −1 SetD∆ .”
Higher Segal Spaces (2012, background): “By a result of Dugger [Dug01], any combinatorial model category is Quillen equivalent to a combinatorial simplicial model category.”
Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, background): “Although any combinatorial model category is Quillen equivalent to a simplicial model category [15], this replacement process can be inconvenient.”