Introduces tidy maps for left exact monoidal localizations and proves both Goodwillie and Weiss towers are generated by them and are completion towers in topoi.
On bi-enriched∞-categories
6 Pith papers cite this work. Polarity classification is still indexing.
6
Pith papers citing it
abstract
We extend Lurie's definition of enriched $\infty$-categories to notions of left enriched, right enriched and bienriched $\infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $\infty$-categories and share many desirable features with them. We use bienriched $\infty$-categories to endow the $\infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $\infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we construct enriched Kan-extensions from operadic Kan-extensions, compute the monad for enriched functors, prove an end formula for morphism objects of enriched $\infty$-categories of enriched functors and a coend formula for the relative tensor product of enriched profunctors and construct transfer of enrichment from scalar extension of presentably bitensored $\infty$-categories. In particular, we develop an independent theory of enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories.
fields
math.AT 6representative citing papers
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.
A V-enriched ∞-operad is equivalent to a presentably symmetric monoidal V-module category generated by a ⊗-atomic marking of its colors.
Proves an oriented Street-Roberts conjecture by presenting (∞,∞)-categories as sheaves on families of oriented polytopes, generalizing Campion's work.
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.
citing papers explorer
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Left exact monoidal localizations from tidy maps
Introduces tidy maps for left exact monoidal localizations and proves both Goodwillie and Weiss towers are generated by them and are completion towers in topoi.
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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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Enriched $\infty$-operads as marked algebras
A V-enriched ∞-operad is equivalent to a presentably symmetric monoidal V-module category generated by a ⊗-atomic marking of its colors.
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An Oriented Street--Roberts Conjecture
Proves an oriented Street-Roberts conjecture by presenting (∞,∞)-categories as sheaves on families of oriented polytopes, generalizing Campion's work.
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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.