Search Results

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 …

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 …

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it a $\ma …

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 …

I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and …

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 …

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 …

Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and cons …