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Say $\mathcal{M}$ is a model category and $A$ is a cofibrant object of $\cal{M}$. Under these conditions, there's an undercategory $\mathcal{M}_{A/}$ whose objects are pairs of an object $X$ of $\math …

This is false for spaces. Let $X = S^0, Y = S^1$, and $f:X \to Y$ be the trivial map. Then $Z = Cf$ is $S^1 \vee S^1$. Then $[X,Y]$ is trivial, so then the the truth of this statement would imply: …

Stefan Schwede's "An unititled book project about symmetric spectra" covers, in chapter III, the projective levelwise and stable model structures on symmetric spectra in quite some detail (along with …

Under the injective model structure or the Reedy model structure, the answer is yes: this is a left Bousfield localization. For any $X \in M$, define $\Delta[n] \otimes X$ to be the element of $M^\De …

The core problem is: Maps of symmetric spectra are, levelwise, maps equivariant with respect to the symmetric group, but the notion of weak equivalence ignores that. The injectivity notion essentiall …

The cotangent complex $\Bbb L_{B/A}$ is a (homologically) bounded-below complex of projective $B$-modules, and the bottom homology group is $H_0 (\Bbb L_{B/A}) = \Omega_{B|A}$. If we apply $RHom_{B-m …

Dear Aaron, For question 1, you're correct in that all objects are cogroup objects. This isn't necessary for the original version from Bousfield's paper--only one of the suspensions is--and there yo …

Here are two variants on this. Strictly, there is no such model structure. If $X$ is any space, then the map $X_+ \to CX_+$ is an acyclic cofibration under the definitions given, where $CX$ is the co …

The model structure on this category has a set of generating cofibrations: if we write $F$ for the free cdga functor, $k$ for the complex with value $k$ concentrated in degree zero, and $I$ for the ma …

Here is a short argument why we don't expect generalized cohomology theories to behave so well. In the stable homotopy category, there is a generalized homology/cohomology theory represented by the s …

This is an elaboration on Lennart's comment. This can be made to come from a Quillen equivalence. Here are the ingredients you'd usually need to show it. (Sorry, I don't have my copy of EKMM handy …

The two categories you describe are not equivalent in the fashion that you hope. No matter what kind of simplicial category $C$ is, the quasicategory $N_\Delta(C)$ has an explicit description of its h …

Here is a classical example. Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic $p$. Weak equivalences are quasi-isomorphisms, fibrati …

This is just to be explicit about the role of the bar construction in David's answer. If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: …

The answer is that the double complex does not collapse in the case of $HX$ either. The issue is that you have a coherent simplicial diagram valued in chain complexes with differential zero, rather th …