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12

The issue you noticed is an artifact of the fact that May and Ponto seem to have gotten their proof from Cole's 2006 paper "Many homotopy categories are homotopy categories." In Barthel and Riehl's 2013 paper "On the construction of functorial factorizations for model categories," it is noted that Williamson noticed this error, and in (5.5) and §6.1 Barthel ...


10

The grammar is "a category is tensored over a monoidal category"; this is a generalization of a set being equipped with an action of a monoid, or an abelian group being equipped with an action of a ring. In full generality you should provide the tensoring, but sometimes if you require enough it exists uniquely. The general pattern of the uniqueness results ...


8

This is in many places. I think that the whole point of model categories is that people saw you could get a lot of mileage out of the adjunction between Top and sSet (geometric realization and singular simplices). Just how good is the combinatorial model for topological spaces? The answer is that it is a Quillen equivalence (whatever that means). But for ...


6

My impression is that a "homotopy theory" is more than the homotopy category: it is an $(\infty, 1)$-category. The paper you describe introduces one model for $(\infty, 1)$-categories (complete Segal spaces); the most popular one today seems to be the "quasi-categories" used by Lurie and Joyal. A slogan is that model categories are presentations for ...


6

Apart from the two factorization axioms, the proof in Dwyer-Spalinski's section 7 goes through in the desired generality and it doesn't make a detour via Dold-Kan. Therefore I'll only fill in the part of the argument depending on the small object argument in all the sources I know. In order to produce factorizations without the small object argument, you ...


5

Let $M$ be a chain complex of $R$-modules, concentrated in degrees $\ge 0$. Define a chain complex $\textrm{Cyl}(M)$ as follows: $$\textrm{Cyl}(M)_n = M_n \oplus M_{n-1} \oplus M_n$$ $$\partial(a, b, c) = (\partial a + b, - \partial b, \partial c - b)$$ There are evident chain maps $i_0, i_1 : M \to \textrm{Cyl}(M)$ and $p : \textrm{Cyl}(M) \to M$: ...


4

This is essentially the same answer as Zhen Lin, but I will offer a different point of view. We may define an interval object ,$\mathcal{I}$ in the category of chain complexes over $R$. We will define the zero degree R- module as $R[x,y]$ and the degree one R-moldule as $R[I]$, and all other degrees will be zero. The boundary map will be $$ \partial ...


4

A zig-zag of natural equivalences from a functor $F$ to a functor $G$ means a sequence of functor $F_1,\cdots, F_n$, with $F_1=F$ and $F_n=G$, and natural equivalences (in the context above, these are natural transformations whose components belong to the class of weak equivalences) $\alpha_1 ,\cdots ,\alpha_{n-1}$ such that, for each $\alpha_i$, either the ...


4

I have a vague memory of being told that someone has proven that the Strom model structure is not cofibrantly generated. I would ask Boris Chorny or Carles Casacuberta. That said, a map $f \colon X \to Y$ is a Hurewicz fibration if and only if it lifts against the inclusion $Nf \to Nf \times I$, where $Nf$ denotes its mapping cocylinder. From this ...


4

I like Philip S. Hirschhorn. Model categories and their localizations. Mathematical Surveys and Monographs 99. Providence, RI: American Mathematical Society, 2003, pp. xvi+457. ISBN: 0-8218-3279-4. MR1944041. First, a word of warning: the book is divided into two parts, but the first part depends logically on the second part. Thus, beginners should ...


4

Pardon the sketchy details to follow here: there are many details that I'm burying under the rug for sake of brevity (and giving some sort of answer!). I'd recommend the first chapter of Hovey's book to get an idea for how these arguments go. A more geometric example of this kind of argument is the construction of homotopy colimits, which are a "derived" ...


4

The definition of $D(\sigma)$ is not correct, but almost, it might be a typo in the book: Say, we have $\mathcal A\overset{F,F'}\to\mathcal B$, then the natural transformation $U\sigma\circ\eta$ goes $1_{\mathcal A}\to UF\to UF'$ while the domain of $U\varepsilon'$ is $UF'U'$. So we are missing an $U'$ here, by syntax, and the correct version would be ...


3

A quick introduction to the model category-theoretic method in homological algebra can be found in Goerss and Schemmerhorn's "Model Categories and Simplicial Methods". They treat resolutions in the nonabelian setting with the language of model categories. A central tool in abstract homotopy theory is the simplicial set, so a good warmup if you've not seen ...


3

Let $\mathbf{A}$ be an abelian (or Grothendieck) category, and consider the injective model structure on the category of cochain complexes $\mathrm{Ch}(\mathbf{A})$. Then the derived category $\mathcal{D}(\mathbf{A})$ is the homotopy category $\mathrm{Ho}(\mathrm{Ch}(\mathbf{A}))$ by inverting the weak equivalences.


3

When people say "unique up to unique isomorphism" it must always be understood in the appropriate sense. This particular case is an instance of the uniqueness of objects defined by adjunctions. Suppose we have an adjunction $$F \dashv U : \mathcal{D} \to \mathcal{C}$$ where $F : \mathcal{C} \to \mathcal{D}$ is the left adjoint. Then, for each object $C$ in ...


3

Thank you for your responses. Based on the discussion, the following conventions should be consistent and convenient: The set of horn inclusions is the set $\{ \Lambda^n_k \to \Delta^n\ |\ n \geq 1,\ 0 \leq k \leq n \}$. This is the set of generating acyclic cofibrations for the standard model structure on (unpointed) simplicial sets. A map $f: X \to Y$ of ...


3

As Aaron observes in the comments, this is in Strøm's "The homotopy category is a homotopy category."


3

The meaning of "functorial factorization" is well clarified in the following article http://arxiv.org/pdf/1204.5427.pdf at page 7. What you require is the existence of two factorization functors $R, L : \mathrm{Mor}(M) \to \mathrm{Mor}(M)$ which can be used to factor any morphism, and are well behaved with respect to domains and codomains. That is: ...


2

Maybe you're beginning your journey through model, localized, homotopy categories by a steep way. I would try this short paper first: W. G. Dwyer and J. Spalinski.


2

The usual notion of equivalence for model categories is due to Quillen: given model categories $\mathcal{M}$ and $\mathcal{N}$, an adjunction $$F \dashv G : \mathcal{N} \to \mathcal{M}$$ is a Quillen equivalence if it is a Quillen adjunction (i.e. $F$ preserves cofibrations and trivial cofibrations, and $G$ preserves fibrations and trivial fibrations) such ...


2

A long comment: When you have a model category $\mathcal{M}$ in particular you have an enriched $Ho(sSet)$-module structure on $Ho(\mathcal{M})$ (in the sense of Hovey) i.e. the homotopy category $Ho(\mathcal{M})$ is tensored and cotensored over $Ho(sSet)$ in a compatible way. Let me use the following analogy: suppose that $R$ is a ring and $M$ is an abelian ...


2

The generating cofibrations are the inclusions $\emptyset \hookrightarrow \{ \ast \}$ and $\{ 0, 1 \} \hookrightarrow \{ 0 \to 1 \}$ plus the projection $\{ 0 \rightrightarrows 1 \} \to \{ 0 \rightarrow 1 \}$. It is easy to see that a functor has the right lifting property with respect to these if and only if it is fully faithful and surjective on objects.


2

There is a map from $S^1$ to the pseudocircle which is a weak equivalence, but not a homotopy equivalence. In fact, because $S^1$ is Hausdorff there are no nonconstant functions from the pseudocircle to $S^1$. This answers the second question positively; the range being a CW-complex is important to being able to construct a homotopy inverse in general. I ...


2

W. G. Dwyer and J. Spaliński. “Homotopy theories and model categories”. In: Handbook of algebraic topology. Amsterdam: North-Holland, 1995, pp. 73–126. DOI: 10.1016/B978-044481779-2/50003-1. MR1361887. This paper is an introduction to the theory of model categories. The prerequisites needed to read it are very limited, and most (if not all) the proofs ...


2

You could take $Y=\text{C}X$ the cone over $X$ and define your square to be cocartesian, i.e. as the (homotopy) pushout of $\text{C}X\leftarrow X\to\ast$. Then $\varphi: \text{cof}(f)\to\text{cof}(g)$ is an isomorphism since cocartesian squares glue.


2

First, note that $\emptyset \hookrightarrow \Delta^n$ is a cofibration (because any monomorphism is a cofibration in $\textbf{sSet}$), so the right lifting property of an acyclic fibration $p : X \to Y$ implies $p_n : X_n \to Y_n$ must be a surjection. In general, if you know a Kan fibration $p : X \to Y$ restricts to a surjection $p_0 : X_0 \to Y_0$, then ...


2

The geometric realization of a Kan fibration is a Serre fibration, and these are certainly surjective (unless we're in the trivial case that the total space is empty, or more generally the preimage of some path component of the target is empty). But it shouldn't be hard to see directly that a Kan fibration $f:X \rightarrow Y$ must be levelwise surjective, ...


2

This isn't a geometric approach, but for intuition, I prefer the free resolution to the projective resolution. I tend to think of "projective" as being essentially, "All the properties we need in this category to ensure that we get the same results as we would from a free resolution." :) If $...F_3\rightarrow F_2\rightarrow F_1\rightarrow M$ is your free ...


2

There is a common explanation. As you already mentioned, in order to show that pullbacks preserve fibrations one does not use all the axioms of the model category structure but only the lifting properties. Given two maps $f\colon A\to B$ and $g\colon X\to Y$, write $f \mathrel{\Box} g$ if every square $$ \matrix{ A & \longrightarrow & X \cr ...


2

All the relevant structure can be transported along equivalences. Choose a quasi-inverse $G : \mathcal{C} \to \mathcal{M}$ and a natural isomorphism $\epsilon : F G \Rightarrow \mathrm{id}$. Then given a functorial factorisation system $(L, R)$ on $\mathcal{M}$, we can define a functorial factorisation system $(L', R')$ on $\mathcal{C}$ by putting $L' f = F ...



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