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There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ fo …

All of these examples involve a bit of cleverness, so I thought I'd point out a more straightforward way to construct counterexamples. If $X$ is any space, we can build a space $X' = K(\pi_0 X, 0) \t …

I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak Hausdor …

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 nth stage of the Whitehead tower of X is the homotopy fiber of the map from X to the nth (or so) stage of its Postnikov tower, so you can use your functorial construction of the Postnikov tower pl …

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but could there be differentials or extens …

That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left …

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 …

I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×hZ Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets. Assume for con …

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 …

$\require{AMScd}$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 …

Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an equivalen …

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 …

It sounds like maybe you can prove your first statement for simply connected spaces. In that case, you can use the homotopy orbit spectral sequence (the Serre spectral sequence associated to the fibr …

More generally, if A and X0 are any finite CW complexes, and f : A → X0 is any map, let Y be the mapping cone of f, and let X be Y with the cone point removed; then Y is the one-point compactification …