On spectra and affine strict polynomial functors

We compare derived categories of the category of strict polynomial functors over a finite field and the category of ordinary endofunctors on the category of vector spaces. We introduce two intermediate categories: the category of $\mathrm{\infty }$–affine strict polynomial functors and the category of spectra of strict polynomial functors. They provide a conceptual framework for compuational theorems of Franjou–Friedlander–Scorichenko–Suslin and clarify the role of inverting Frobenius morphism in comparing rational and discrete cohomology.