Is there a name for an $\infty$-category C that admits a zero object and suspensions such that the suspension functor $C \to C$ is an equivalence but which does not necessarily admit cofibers and fibers?
Such $\infty$-categories are precisely the full subcategories of some stable $\infty$-category containing the zero object and closed under loops and suspensions. Moreover such $\infty$-categories admit a canonical enrichment in spectra and one can prove there is a canonical equivalence between such $\infty$-categories and spectral $\infty$-categories that admit loops and suspensions.