# Tagged Questions

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ... 1answer 336 views ### Model structure for cooperads and for coalgebras I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ... 2answers 233 views ### Analogues of 'cone' distinguished triangles for pointed model categories? For an additive A and any morphism f:X\to Y in C(A) one has the following distinguished triangle in the homotopy category K(A): X\to Y\to Cone(f)\to X[1]. What is the closest analogue of ... 0answers 148 views ### Jardine model structure as left Bousfield localization This should be a really basic question, but I'm stuck on it. The question. I see written everywhere (for example here, or in the article [DHI] Hypercovers and simplicial presheaves of Dugger, ... 0answers 156 views ### Homotopy theory of acyclic categories Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ... 1answer 281 views ### The state of the art in the rectification of homotopy-coherent structures 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 ... 1answer 229 views ### Quillen equivalence of diagram categories Let C be a model category and B a direct category. By Theorem 5.1.3 in Mark Hovey's book Model categories, there is a model category structure on the diagram category C^B such that weak ... 1answer 184 views ### Bousfield localization before and after taking homotopy The ncatlab says: Under suitable conditions it should be true that for C a model category whose homotopy category \mathrm{Ho}(C) is a triangulated category the homotopy category of a left ... 5answers 579 views ### What is a good basic reference on model categories? I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ... 1answer 122 views ### Equivariant versus retractive spaces: a reference request Let T be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let G = |G.| be the ... 1answer 470 views ### Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors. Is ... 3answers 432 views ### Monoidal model category structure on a functor category. Let A be a small simplicial category. The category Fun(A,s\mathrm{Set}) of simplicial functors from A to simplicial sets can be given the projective model structure in which fibration and weak ... 1answer 147 views ### Reedy model structures on oplax limits Suppose R is a category and F:R\to Cat is a functor (or pseudofunctor). The oplax limit of F is the category whose objects consist of an object x_r \in F(r) for all r together with a ... 0answers 174 views ### relationships between properties of model categories I've recently found myself running up against all sorts of adjectives that can describe a model category: cofibrantly generated, combinatorial, tractable, stable, locally (finitely) presentable, (left ... 1answer 366 views ### When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action? Let M be a closed symmetric monoidal model category. Let X be a cofibrant object (it can also be fibrant if you like) and let \Sigma_n act on X^{\otimes n} by permuting the factors (note that ... 1answer 258 views ### geometric realization on \mathbf{sTop} Is geometric realization |\cdot|:\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}\rightarrow \mathbf{Top} a left Quillen functor? If so, under what model structure on ... 1answer 316 views ### Is the model category of Complete Segal Spaces right proper? Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ... 3answers 538 views ### A reference for Calculus of Functors for Model Categories I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories. I am at least aware of some of the extended ... 0answers 72 views ### Reference for model structure on CosimplicialAbelianGroups There is a standard (simplicial) model category structure on the category Ab^\Delta \simeq Ch^\bullet_+(Ab) of co-simplicial abelian groups, whose fibrations are the degreewise surjections (and weak ... 1answer 362 views ### How to localize a model category with respect to a class of maps created by a left Quillen functor Let M and N be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair$$ L: M\rightleftarrows N: R. I want the following result: There ...
Suppose I have a small category $\mathcal{C}$ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightarrow \mathcal{I}$ ...