One can show that spectrifications of maps between $ \Sigma$ -cofibrant prespectra that are spacewise homotopy equivalences are homotopy equivalences of spectra. I’m interested in the following:
Does this result hold if only the source prespectrum is assumed to be $ \Sigma$ -cofibrant?
The reason for my interest is the following. The adjoint from prespectra to inclusion prespectra is difficult to deal with. I see that sometimes people replace a prespectrum by a thickened one, via a cylinder construction, as in Chapter X, Section 5 of Elmendorf-Kriz-Mandell-May’s “Rings, modules and algebras in stable homotopy theory”, to get a $ \Sigma$ -cofibrant prespectrum so that we have a nicer description of the spectrification. This is done, for example, when defining the topological Hochschild spectrum in Hesselholt-Madsen’s “On the $ K$ -theory of finite algebras over Witt vectors of perfect fields”. Say the original prespectrum is $ t$ , the cylinder construction yielding a $ \Sigma$ -cofibrant prespectrum is denoted by $ K$ and spectrification is denoted by $ L$ . We have a map of prespectra $ Kt \to t$ which is a spacewise homotopy equivalence. I would hope that $ LKt$ and $ Lt$ are related in a nice way so as to say that the intermediate thickening didn’t essentially change anything. The “nice way” I’m hoping for is that one can deduce that the induced map $ LKt \to Lt$ is a homotopy equivalence (or perhaps weak homotopy equivalence).
✓ Extra quality
ExtraProxies brings the best proxy quality for you with our private and reliable proxies
✓ Extra anonymity
Top level of anonymity and 100% safe proxies – this is what you get with every proxy package
✓ Extra speed
1,ooo mb/s proxy servers speed – we are way better than others – just enjoy our proxies!
USA proxy location
We offer premium quality USA private proxies – the most essential proxies you can ever want from USA
Our proxies have TOP level of anonymity + Elite quality, so you are always safe and secure with your proxies
Use your proxies as much as you want – we have no limits for data transfer and bandwidth, unlimited usage!
Superb fast proxy servers with 1,000 mb/s speed – sit back and enjoy your lightning fast private proxies!
99,9% servers uptime
Alive and working proxies all the time – we are taking care of our servers so you can use them without any problems
No usage restrictions
You have freedom to use your proxies with every software, browser or website you want without restrictions
Perfect for SEO
We are 100% friendly with all SEO tasks as well as internet marketing – feel the power with our proxies
Buy more proxies and get better price – we offer various proxy packages with great deals and discounts
We are working 24/7 to bring the best proxy experience for you – we are glad to help and assist you!