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In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{Post}(\mathc …

Here is a simple counterexample with $X = Y = \mathbb{R}$: Send a simplex $\sigma : |\Delta^n| \to \mathbb{R}$ to the affine function $F(\sigma) : |\Delta^n| \to \mathbb{R}$ with the same values at th …

A cofibration category is saturated if it satisfies the following equivalent conditions: Every map which becomes an isomorphism in the homotopy category is already a weak equivalence. The weak equiva …

A somewhat general statement along these lines would be as follows. Suppose $\mathcal{D}$ is a cartesian closed locally presentable category and $d : \Delta \to \mathcal{D}$ is a cosimplicial object w …

$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in Kapulkin-Lumsdaine) is that "the base of the universal fibration is fibrant". Exp …

Pick pointed topological spaces $A$ and $B$ which admit pointed injective continuous maps $A \to B$ and $B \to A$ for which $A$ is contractible but $B$ has nonvanishing reduced homology. For example, …

A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the stru …

The general statement is that if $V$ is a monoidal model category (I will assume the unit object of $V$ is cofibrant, so that there are no funny extra axioms related to the unit) and $M$ is a $V$-mode …

When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\ma …

Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on $\mathca …

The connected oriented surface $\Sigma_g$ of genus $g \ge 1$ is a $K(G_g, 1)$ where $G_g$ has a well-known presentation. For a general group $G$, the mapping space $Map(K(G, 1), K(G, 1))$ has homotop …

1 is doubly wrong. First, you need to distinguished generalized cohomology theories and reduced generalized cohomology theories. If you want to work with the latter, you should replace "a point" in …

[Removed a paragraph relating to an earlier version of the question] You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inve …

Mostly I refer you to my answer here and also this question. To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a m …

There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over Δop). If $F : I \to M$ is a diagram in a model category, one has $$\operatorname{hocolim}_I …