Questions tagged [model-categories]
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
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Model category structure on Set without axiom of choice
There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...
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What are the advantages of various "models" for the motivic stable homotopy category
People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
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Examples of differential cohomology in cohesive $\infty$ topos
I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.
The formulation of differential cohomology in ...
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When is the category of pro-objects a homotopy category?
For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by
$$
{\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm Hom}(...
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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 ...
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Homotopy transfer in the opposite direction
Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
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sSet-enriched categories, quasi-categories and the model-independent theory
sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual ...
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Quasicategories for non-simplicial model categories
If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...
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The weak equivalences in the covariant model structure
Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:
The cofibrations are the ...
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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 ...
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Geometric Meaning of Different K-theories
There is a whole collection of algebraically defined K-theories. My understanding is that algebraic K-theory is a presheaf of spectra $K$ on $\textbf{Sch}/S$ such that the homotopy groups of $K(X)$ ...
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Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
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Example of non accessible model categories
By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...
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What is a model category from an $\infty$ point of view?
A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example:
Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense ...
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Model for the (infinity,1)-category of (homotopy-)limit preserving functors
I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...
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Non-Cartesian Monoidal Model Structure on a Slice Category
Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
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Weak complicial sets: Are the morphisms too strict?
In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
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Do homotopy groups "always" commute with filtered colimits?
It is well-known that homotopy groups, of, say, simplicial sets, commute with filtered colimits.
However, I could not find a reference for an analogous result for homotopy groups of spectra, or, ...
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On diagrams in model categories and rectification
For a model category $C$, I'm denoting $h_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the ...
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Does the right adjoint of a Quillen equivalence preserve homotopy colimits?
Call a diagram $E$ in a model category a homotopy colimit diagram if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A homotopy colimit is defined as the categorical ...
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Why is every object cofibrant in an excellent model category?
In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
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Model Structure/Homotopy Pushouts in topological monoids?
Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.
Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
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What is the "universal problem" that motivates the definition of homotopy limits/colimits (and more generally "derived" functors)?
The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan ...
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Categories on which one can determine all model structures?
Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
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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 ...
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Fibrant-cofibrant models of Eilenberg-MacLane spectra
There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...
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Examples of non-proper model structure
I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
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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 ...
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When did the Joyal model structure on simplicial sets originate?
Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...
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On model categories where every object is bifibrant
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one ...
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Constructing a "geometric" model structure on Cat by localizing the "categorical" model structure
Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...
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Non standard (?) model category structure on (co)chain complexes.
Let $\cal{A}$ be an abelian category with enough projectives and $\mathbf{C}_+ (\cal{A})$ the category of bounded below chain complexes.
Since Quillen (Homotopical algebra, 1.2, examples B), there is ...
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Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?
Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
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Model category structures on categories of complexes in abelian categories
Section 2.3 of Hovey's Model Categories book defines a model category structure on Ch(R-Mod), the category of chain complexes of R-modules, where R is a ring. Lemma 2.3.6 then essentially states (I ...
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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 ...
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Property-like structure in a model category
In a model category, I have tools to show that mapping spaces are contractible. But if I want to show a mapping space is empty or contractible, is there anything I can do on general grounds?
The idea ...
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global fibrations of simplicial sheaves
I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : ...
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Simple question: different definitions of Bousfield localization
I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
First definition: Let $\mathbf{...
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Schemes as a model category
I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.
A natural temptation is to relate this machinery to ...
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Is the Grothendieck construction a homotopy pullback?
The category of elements of a functor $F:\mathcal C\to\mathsf{Set}$ can be obtained as the strict pullback in with the forgetful functor of pointed sets $\mathsf{Set_*}\to\mathsf{Set}$:
$$
\begin{...
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What is the right notion of a functor from an internal topological category to a topologically enriched category?
Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:
What is the correct notion of a ...
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Which statements and arguments of Hovey's "Model categories" fail without functorial factorizations of morphisms?
I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations ...
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A category with weak equivalences that is not a model category
I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{...
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Example of a CW complex not homeomorphic to the realization of a simplicial set?
I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
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When do the Reedy and injective model category structures agree?
Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on $\...
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The cofibration/fibration $\leftrightarrow$ epi/mono confusion
A recent question Why do we need model categories? reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I ...
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What are the advantages of simplicial model categories over non-simplicial ones?
Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
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Are cofibrations accessible?
The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable ...
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Model structure on the category of small $A_\infty$ categories, hocolims.
I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...
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The definition of Reedy category
The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
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