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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 ...

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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 ...

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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 $p : \textrm{Cyl}(M) \to M$: ...

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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 $$\partial ... 3 The meaning of "functorial factorization" is well clarified in the following article http://arxiv.org/pdf/1204.5427.pdf at page 7. What you require is the existence of two factorization functors R, L : \mathrm{Mor}(M) \to \mathrm{Mor}(M) which can be used to factor any morphism, and are well behaved with respect to domains and codomains. That is: ... 3 When people say "unique up to unique isomorphism" it must always be understood in the appropriate sense. This particular case is an instance of the uniqueness of objects defined by adjunctions. Suppose we have an adjunction$$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 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 ... 3 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" ... 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 Deterministic-Static-Discrete: Clock cycles for a computer program to run on a given input. Deterministic-Static-Continuous: Amount of fluid a pipe can hold before breaking. Deterministic-Dynamic-Discrete: CPU percentage upon startup Deterministic-Dynamic-Continuous: Arguably everything part of the classical physical model Stochastic-Static-Discrete: ... 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 ... 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 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 Given a natural transformation of functors \mathcal{C}' \to \mathcal{C}', say h : f r \Rightarrow \mathrm{id}_{\mathcal{C}'} and a functor f : \mathcal{C} \to \mathcal{C}', we can form a natural transformation h f : f r f \Rightarrow f by taking the components of h f at an object c in \mathcal{C} to be the component h_{f c}. You can view all ... 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 ... 2 The obvious categorical structure, in my view, is to put in a morphism (C, W, F) \to (C', W', F') if and only if the identity functor is a right Quillen functor from (C, W, F) to (C', W', F'), or more simply, if and only if F \subseteq F' and F \cap W \subseteq F' \cap W'. Then: The terminal object is the model structure where C = \{ ... 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}$$is a Quillen equivalence if it is a Quillen adjunction (i.e. F preserves cofibrations and trivial cofibrations, and G preserves fibrations and trivial fibrations) such ... 2 Homotopy equivalent to the terminal object? But I don't know how useful this could be. 2 There is a common explanation. As you already mentioned, in order to show that pullbacks preserve fibrations one does not use all the axioms of the model category structure but only the lifting properties. Given two maps f\colon A\to B and g\colon X\to Y, write f \mathrel{\Box} g if every square$$ \matrix{ A & \longrightarrow & X \cr ...

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The assumption that the $X_\alpha$ form a $\lambda$-sequence includes in particular that for every limit ordinal $\kappa \leq \lambda$ we have $X_\kappa = \varinjlim_{\alpha \lt \kappa} X_\alpha$. Since directed colimits are unchanged by passing to cofinal subsets of the indexing set, we have $X_\mu = \varinjlim_{\alpha_n} X_{\alpha_n}$. In other words, ...

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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 ...

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The following calculation shows the claim using just vertical composition of morphisms (horizontal one only implicitly): \begin{align*} (D^2\sigma)_X&=\varepsilon_{F'X}\circ F(D(\sigma)_{F'X})\circ F\eta'_X &\text{by definition}\\ &=\varepsilon_{F'X}\circ F(U\varepsilon'_{F'X}\circ U\sigma_{U'F'X}\circ \eta_{U'F'X})\circ F\eta'_X&\text{by ...

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Have a look on Joyal's page on model categories. In particular Corollary 1 shows that if the model category $E$ is locally small than the associated homotopy category is also locally small. I guess, as you say, in practice this is not of much concern since the catgeories we are usually concerned with are locally small

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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 ...

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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.

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The claim is a special case of Example 23.8 of [Shulman, Homotopy limits and colimits and enriched homotopy theory].

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The definition you're giving might be right, but it's obscuring the point. When you're replacing $f$ with a fibration, what you're really doing is making it so that the honest pullback of the new square is the homotopy limit of the old square. The usual definition of "homotopy cartesian square" is just that the map from the initial vertex to the homotopy ...

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