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12
votes
1answer
512 views

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 ...
3
votes
2answers
192 views

Free Monoids in Closed Symmetric Monoidal Categories

There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated. Give some category $\mathcal{C}$ with the ...
3
votes
2answers
197 views

Fitting desired weak equivalences and cofibrations into a model category

Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and $\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are the weak equivalences and the ...
-1
votes
2answers
583 views

Alternative characterization of homotopy equivalence

Using the formalism of model categories its possible define the concept of homotopy as done here. If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and ...
8
votes
1answer
477 views

Hovey's unit axiom in monoidal model categories

Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...
4
votes
2answers
282 views

Fibration of Batanin/Leinster $\omega$-groupoids

Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster? I tried to search on Google and in Higher Operads, Higher Categories of Tom ...
4
votes
2answers
472 views

Is the category of small categories locally presentable?

I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...
10
votes
5answers
1k views

Computations in $\infty$-categories

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...
4
votes
4answers
326 views

Find weak equivalences from fibrations and cofibrations

Let's suppose I have a category $\mathcal C$ with two weak factorisation systems $(C,F_W)$ and $(C_W,F)$, where $C_W\subset C$ and $F_W\subset F$. I would like to have a model structure on $\mathcal ...
6
votes
1answer
738 views

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 ...
5
votes
1answer
339 views

Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...
5
votes
0answers
194 views

Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...
3
votes
3answers
389 views

Homotopic maps out of cofibration sequences

Consider a model category $\mathcal{C}$ and a sequence of cofibrations $0 \to X_0 \to X_1 \to X_2 \to \dots$ lying in $\mathcal{C}$. Let $X$ be the colimit of this sequence. Suppose furthermore that ...
7
votes
2answers
535 views

Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. The real question here is, do simplicial object in a ...
1
vote
0answers
138 views

Can one recover an A-algebra from its cotangent complex?

Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor ...
32
votes
4answers
4k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
4
votes
0answers
361 views

New model Structure on $E_{\infty}$-algebras?

Let $\mathbf{sSet}$ be the category of simplicial sets. Is it possible to put a new model structure on $\mathrm{E}_{\infty}$-algebra (of simplicial sets) such that the weak equivalences and fibrations ...
3
votes
2answers
188 views

A model structure on the category of “dualizing maps”

Let $C$ be the category with objects being maps $h:M\to A$ where $A$ is a commutative graded $\mathbb{Q}$-algebra (cdga), $M$ is a differential graded (dg) $A$-module and $h$ is an $A$-dg-module map, ...
12
votes
2answers
1k views

Acyclic models via model categories?

Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation ...
9
votes
1answer
374 views

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 ...
4
votes
0answers
215 views

When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...
4
votes
3answers
751 views

univalent axiom as a property of a model category?

I am interested to understand the univalence axiom of Voevodsky; however, I know very little type theory. Thanks to response below, I now understand what is being univalent means for a morphism. A ...
4
votes
0answers
280 views

Equivariant sheaves and simplicial varieties

I would like to proof the following theorem: Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on ...
4
votes
0answers
351 views

Fiber sequences in proper model categories

I am confused about the notion of a fiber sequence (or dually a cofiber sequence) in a general pointed and proper model category $\mathcal{C}$. Following Hovey, we can define, like in topology, a map ...
4
votes
0answers
448 views

When are “diagrams of cofibrations” projectively cofibrant?

Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a diagram of cofibrations if for every object $p\in P$, $F(p)$ is ...
3
votes
0answers
167 views

Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
2
votes
1answer
314 views

Resolutions by Adapted Class of Objects and Model Categories

My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) ...
4
votes
0answers
166 views

Mapping into a geometric realization.

Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map \[ f: \text{Sing}\,X\to S. \] When is is possible to produce a map $X\to |S|$? We can take ...
9
votes
1answer
729 views

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 ...
4
votes
1answer
276 views

Does adding degeneracies to a semi-simplicial diagram change the homotopy colimit?

Let $\Delta_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta_{+} \hookrightarrow ...
2
votes
0answers
135 views

“non-Bousfield” localisations of model categories

When can I localise a model category by a set(or class) of morphisms, and how do I describe the localised model category ? By 'localise' I mean to find a localisation functor $q_S : M \rightarrow ...
1
vote
0answers
283 views

Is there a model category structure on non-negatively graded commutative chain algebras?

Let $\mathtt{DGA}$ be the category of non-negatively graded DG chain algebras, and $\mathtt{DGA}$* the category of non-negatively graded cochain algebras. Let $\mathtt{CommDGA}$ be the full ...
2
votes
0answers
287 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
4
votes
2answers
306 views

Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...
1
vote
1answer
141 views

Cofibrations and coequalisers in a proper model category

I have a proper model category and in it two coequalisers, $A_i \rightrightarrows B_i \to C_i$, $i=1,2$. I have a map of diagrams arising from maps $A_1 \to A_2$, $B_1 \to B_2$ where these two arrows ...
12
votes
2answers
631 views

Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
1
vote
0answers
130 views

end of a weak equivalence

I would like to get a concrete description of sufficient conditions for the end of a morphism in $\mathcal{C}^{J^{op}\times J}$ (which is a point-wise weak equivalence) to be a weak equivalence. In ...
3
votes
1answer
254 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 ...
2
votes
1answer
280 views

Model categories and cellular maps

A question came up on MSE and it generated, for me, the following question: When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
10
votes
1answer
898 views

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, ...
3
votes
2answers
485 views

Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question. Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...
6
votes
1answer
465 views

When does a cosimplicial object compute homotopy colimits?

Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric ...
15
votes
4answers
1k views

How canonical is cofibrant replacement?

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...
8
votes
2answers
660 views

“Strøm-type” model structure on chain complexes?

Background The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...
3
votes
2answers
326 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
6
votes
0answers
581 views

An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point: Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with the contravariant model structure (cofibrations are monomorphisms ...
4
votes
1answer
229 views

Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category $\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...
5
votes
1answer
744 views

Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...
11
votes
2answers
472 views

Pointed Hurewicz model structure

In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and ...
19
votes
4answers
1k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...