# Tagged Questions

Model categories are categories with three distinguished classes of morphisms: the weak equivalences, the fibrations and the cofibrations. They provide a natural setting for (homotopy-theory) in an arbitrary category, by mimicking the usual properties of (co)fibrations and weak equivalences in ...

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### Weak equivalence iff isomorphism in homotopy category?

I know that a weak equivalence becomes an isomorphism in the homotopy category but is the opposite direction true? Suppose we have a map $f: C\rightarrow D$ in a model category. If $f$ becomes an ...
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### Model category that doesn't admit functorial factorizations?

I guess it's a modern convention that model categories are typically required to have functorial factorizations. In the cofibrantly generated case, the factorizations constructed by the small object ...
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### Why is the Quillen model structure so painful to find?

Proving that the category of simplicial sets carries the Quillen model structure is undoubtedly difficult; the book by May and Ponto "A more concise course in algebraic topology" makes a considerable ...
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### Limits in a Model Category

I've become interested in how the axioms of a model category have changed, since originally posed by Quillen; in particular, that Quillen only originally required finite limits and colimits, however ...
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### Categorical way of making monos commute

Is there a categorical way (in terms of diagrams, limits, lifting properties etc) to formulate the requirement that for every pair of monos $f,g:C \to D$ there should be an endomorphism $h:D \to D$, s....
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### Fiber of fibration of simplicial sets

$\require{AMScd}$ If $p:E\to B$ is a fibration of simplicial sets, is the fiber in the model category sense, i.e. the homotopy limit of $$\begin{CD}{} @. E \\@. @VVV \\*@>>> B \end{CD}$$ the ...
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### Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^âˆ—, Y^âˆ—$ and $f : X â†’ Y$ is a morphism in $\mathcal{C}$, then the left dual map $f^âˆ— : Y^âˆ— â†’ X^âˆ—$ of $f$ is given by: ...
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### Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence ${}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of $H$-...
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### Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...
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### If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
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### Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where: weak-equivalences are quasi-isomorphisms; cofibrations ...
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### Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
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### Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $(n - 1)$-sphere in the $n$-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
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### “Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
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### What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley (http://www.math.uni-bonn.de/people/...
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### A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
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### Homotopy split monomorphisms [closed]

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose ...
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### definition of a model category

In Quillen's book ,,Homotopical Algebra" he defines a model category as a category with three types of arrows satyfying some axioms. I have problems with understanding two of them. These axioms says ...
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### Homotopy and chain homotopy determine each other

In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory: Proposition (3.7). [page 210] ...
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### Constructing model category from given category

Given a model category $\mathcal{M}$, Goerss and Hopkins constructed a subcategory (see Structured Ring Spectra, p. 160) $\mathbf{E}$ of $\mathcal{M}$ such that: If $X\in\mathbf{E}$ and $Y$ is ...
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### Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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### Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...
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### Chain complexes as a model category?

I'm reading a paper called Model Categories and Simplicial Methods by Paul Goerss and Kristen Schemmerhorn, and they show that cofibrations have lifting property with respect to acyclic fibrations for ...
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### Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
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### are equivalences in an $(\infty,1)$-category preserved under colimits

Let $C$ be an $(\infty,1)$-category (e.g. quasicategory) having all small colimits. If $f_i:x_i \to y_i$ are equivalences in $C$ indexed by a small set $I$, is $f:=\mathrm{colim}_{i \in I} f_i$ an ...
Is it an open problem if $\mathbf{TOP}$ with Hurewicz (StrĂ¸m) model structure is cofibrantly generated?