External stable enhancement and smash product of spectra

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Let S be the infinity-category of spaces and S_^fin the infinity-category of finite pointed spaces. Let Sp=Exc_(S_*^fin,S), the infinity-category of functors that take the point to a contractible space and pushout squares to pullback squares. Sp is stable and presentable, and is equivalent to the homotopy limit of the tower of pointed spaces under the loop functor Ω. An object of that limit is a coherent tower $Y_n$≃Ω$Y_{n+1}$, and its nth space is Ω∞(Y[n]), where [1] is suspension. Sp has all small limits and colimits; a square is a pushout exactly when it is a pullback. Evaluation Ω∞ has a left adjoint Σ∞, and the sphere spectrum is 𝕊=Σ∞S⁰. Sp has a symmetric monoidal smash product ∧ with unit 𝕊, preserving small colimits separately in each variable.

Given: Spaces and diagrams small in a fixed universe, with a larger universe containing their infinity-categories.

Goal: Supply the stable enhancement, coherent Omega-tower description and smash product.

[F1] Lurie's results at the cited locators give the stated stabilization, tower equivalence, adjunction and monoidal structure under the stated hypotheses.

Import [F1]. An infinity-category retains mapping spaces and coherent higher compositions. A coherent diagram is a functor from an indexing infinity-category; its limits and colimits have universal properties expressed by equivalences of mapping spaces. Stability includes a zero object and identifies pullback squares with pushout squares. The associativity, symmetry and unit laws of smash include their higher coherences. These foundations are external inputs, not locally reconstructed proofs.