I am interested in the Ω-spectrum Xᵢ, Xᵢ ≅ $ΩX_{i+1}$, where $Xᵢ := BⁿX$ for a CW-complex $X$. This construction is left adjoint, right?
It seemed like it wouldn't be very hard to construct the smash product of two Ω-spectra, especially since smash product is supposed to be left adjoint to hom. I've been told that it is either difficult or impossible to construct a functorial smash product for Ω-spectra, which is one of the selling points of other models.
It's well known that there isn't a category of spectra with a [certain five] 1 properties. But this construction with B suggested a different unit.
The construction bears similarities with localization, where we have this:
- Let R be a ring and let x : R be an element. $R[x^{-1}]$ is an R-algebra in which $x$ has an inverse. For each $R$-algebra $S$ in which $x$ is invertible, there is a unique map $R[x^{-1}] ⭢ S$ which produces a commutative triangle.
My question is, is it true that:
- Let *-∞-Grpd be the category of based infinity groupoids and let $S¹ : $∞$-Grpd$ be the one sphere. The construction above involving B gives a left adjoint from *-∞-Grpd to $Ω$-spectra sending wedge product to smash product. For each left adjoint $Φ$ from *-∞-Grpd to a complete monoidal closed category $C$ in which $ΦS¹$ is invertible under tensor up to isomorphism, and sending wedge product to smash product up to isomorphism, there is a unique left adjoint up to natural transformation from Ω-spectra producing a commutative diagram.
The above is to be understood as up to isomorphism and natural isomorphism where appropriate.