Witt vectors and $\delta$-Cartier rings

We give a universal property of the construction of the ring of $p$-typical Witt vectors of a commutative ring, endowed with Witt vectors Frobenius and Verschiebung, and generalize this construction to the derived setting. We define an $\mathrm{\infty }$-category of $p$-typical derived $\delta$-Cartier rings and show that the derived ring of $p$-typical Witt vectors of a derived ring is naturally an object in this $\mathrm{\infty }$-category. Moreover, we show that for any prime $p$, the formation of the derived ring of $p$-typical Witt vectors gives an equivalence between the $\mathrm{\infty }$-category of all derived rings and the full subcategory of all derived $p$-typical $\delta$-Cartier rings consisting of $V$-complete objects.