We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent transferred weak model structures on both the categories of algebraically cofibrant and algebraically fibrant objects. Under additional assumptions, these transferred weak model structures are shown to be left, right or Quillen model structures. By combining both constructions, we show that each combinatorial weak model category is connected, via a chain of Quillen equivalences, to a combinatorial Quillen model category in which all objects are fibrant.
We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by $\mathcal{C}_{fc}$ the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of $\mathcal{C}$ at the weak equivalences can be computed as a bicategory $\mathcal{H}o(\mathcal{C})$ whose objects and arrows are those of $\mathcal{C}_{fc}$ and whose 2-cells are classes of $q$-homotopies up to an equivalence relation. When considered …
We define a homotopy relation between arrows of a category with weak equivalences, and give a condition under which the quotient by the homotopy relation yields the homotopy category. In the case of the fibrant-cofibrant objects of a model category this condition holds, and we show that our notion of homotopy coincides with the classical one. We also show that Quillen's construction of the homotopy category of a model category, in which the arrows are homotopical classes of a single arrow between fibrant-cofibrant objects, can be made as well for categories with weak equivalences using this no…
In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.
Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectr…
We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from Hess, which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a …
Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model structure are not well understood calculationally. To illustrate, we exhibit some examples of cofibrant and fibrant posets and an example of a non-cofibrant finite poset.
We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2.
In this paper, we consider the model structure on the category of cellular sets originally conjectured by Cisinski and Joyal to give a model for the homotopy theory of weak (ω)-categories. We demonstrate first that any (Θ)-localizer containing the spine inclusions (ι: \Sp[t] \hookrightarrow Θ[t]) must also contain the maps (X\times ι: X\times \Sp[t] \hookrightarrow X\times Θ[t]) for all objects ([t]) of (Θ) and all cellular sets (X). This implies in particular that a cellular set (S) is local with respect to the set of spine inclusions if and only if it is Cartesian-local. However, we show tha…
We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.
This paper proves that the q-model structures of Moore flows and of multipointed $d$-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed $d$-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a …
A natural generalization of locally noetherian and locally coherent categories leads us to define locally type $FP_{\infty}$ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in a paper of Bravo-Gillespie-Hovey. We show that $\mathcal{D}(\mathcal{AC})$, the derived category of absolutely clean objects, is always compactly generated and that it is embedded in $K(Inj)$, the chain homotopy category of injectives, as a full s…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by func…
In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $Σ\subset \mathscr{A}$ containing the identities, we construct a 2-category $\mathcal{H}o(\mathscr{A})$ obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in \cite{e.d.2}. The inclusion 2-functor $\mathscr{C} \longrightarrow \mathcal{H}o(\mathscr{A})$ has a universal property which implies that it will be the 2-localization of $\mathscr{A}$ at $Σ$ as soon as the a…
Cotorsion pairs and model structures induced by resolution dimensions relative to Frobenius pairs
Centro de Ciencias Matemáticas. Universidad Nacional Autónoma de México（墨西哥国立自治大学数学科学中心）；Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”. Facultad de Ingeniería. Universidad de la República（乌拉圭共和国大学工程学院拉富尔加里亚教授数学与统计研究所）
A recent result by J. Šaroch and J. Šťov\'ıček asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by …
