Homotopy posets assemble into an oriented long exact sequence analogue and form layers of a categorical Postnikov tower, with Postnikov-complete (∞,∞)-categories identified as the limit of (∞,n)-categories along truncation functors.
The higher algebra of weighted colimits.arXiv: 2406.08925
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Inverting endomorphism categories produces a stable homotopy theory of higher categories in which categorical spectra classify homology theories via a categorical Brown representability theorem.
Defines categorical homology via an Eilenberg-Steenrod analogue, proves a Dold-Kan correspondence using the Street nerve, and derives a Dold-Thom theorem for multiplicative structure and globe computations.
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Homotopy Posets, Postnikov Towers, and Hypercompletions of $\infty$-Categories
Homotopy posets assemble into an oriented long exact sequence analogue and form layers of a categorical Postnikov tower, with Postnikov-complete (∞,∞)-categories identified as the limit of (∞,n)-categories along truncation functors.
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Stable homotopy theory of higher categories
Inverting endomorphism categories produces a stable homotopy theory of higher categories in which categorical spectra classify homology theories via a categorical Brown representability theorem.
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Homology of higher categories
Defines categorical homology via an Eilenberg-Steenrod analogue, proves a Dold-Kan correspondence using the Street nerve, and derives a Dold-Thom theorem for multiplicative structure and globe computations.