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Mike Hopkins tells me it is indeed true that for any A∞ cogroup space Y, the homotopy limit X of the associated cobar complex is a desuspension of Y (Y = ΣX as A∞ cogroups)--we don't even need any con …

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 …

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 …

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 …

By taking K = a simplex and L = its boundary you can show that |A| -> |X| is an isomorphism on all homotopy groups (do surjectivity and injectivity separately). Then apply Whitehead's theorem.

The cohomology ring is representable by the product of all the Eilenberg-Mac Lane spaces K(Z, n) as n varies. Note that this gives the product of the abelian groups HnX, not their direct sum. For in …

Even if you don't ask for the ring structure to be preserved, this seems quite unlikely, at least if you require the functor to do the same "scaling up" operation to maps on cohomology induced by maps …

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 …

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 …

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 …

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 …

You might want to take a look at the responses to How to think about model categories? Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical …

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 …

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 …