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Keywords:
Quillen cohomology; ($\infty,2$)-category; tangent category; spectrum; Grothendieck construction
Quillen cohomology; ($\infty,2$)-category; tangent category; spectrum; Grothendieck construction
Summary:
In this paper we study the homotopy theory of parameterized spectrum objects in the $\infty$-category of $(\infty,2)$-categories, as well as the Quillen cohomology of an $(\infty,2)$-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an $(\infty,2)$-category $\Bbb C$, which we call its twisted 2-cell $\infty$-category. We then establish an equivalence between parameterized spectrum objects over $\Bbb C$, and diagrams of spectra indexed by the twisted 2-cell $\infty$-category of $\Bbb C$. Under this equivalence, the Quillen cohomology of $\Bbb C$ with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between $(\infty,1)$-categories are uniquely determined at the level of the homotopy (3,2)-category of Cat$_\infty$.
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