Cauchy-completions and higher fusion categories

Abstract

Fusion categories provide algebraic input for three-dimensional topological field theories and arise throughout quantum algebra, representation theory, and low-dimensional topology. This thesis develops foundations for their higher-categorical analogues using enriched ∞-category theory. We characterize the Morita n-category of separable multifusion (n−2)-categories by a universal property as a higher Cauchy-completion of vector spaces. We further identify its objects as precisely the fully dualizable objects in the Morita n-category of monoidal (n−2)-idempotent-complete k-linear (n−2)-categories. By the Cobordism Hypothesis, these objects therefore determine fully extended framed topological field theories.

The first part gives a model-independent description of enriched ∞-categories. For a presentably monoidal ∞-category 𝒱, we prove that a 𝒱-enriched ∞-category is determined by its enriched presheaf category together with a marking by its representable presheaves. More precisely, we construct an equivalence between 𝒱-enriched ∞-categories and a certain full subcategory of presentable 𝒱-module ∞-categories equipped with a map from an ∞-groupoid. This perspective reduces many aspects of the enriched theory to the formalism of presentable ∞-categories. We use it to describe univalence and tensor products of enriched ∞-categories, and later develop weighted colimits and profunctors within the same framework.

The second part extends this approach to 𝒱-enriched ∞-operads. We describe them as marked presentably symmetric monoidal 𝒱-module ∞-categories and relate this description to their presentation as algebras in symmetric sequences. For enrichment over ∞-groupoids, we compare our construction with Lurie's ∞-operads and prove that the corresponding ∞-categories of algebras agree. We also study operadic presheaves, envelopes and free algebras.

The final part develops the theory of Cauchy-complete enriched ∞-categories and extends it iteratively to (∞,n)-categories. Depending on the enrichment, Cauchy-completeness for instance recovers the concepts of idempotent-completeness, additivity, or stability. Iterated Cauchy-completion yields a notion of higher idempotent-completeness that agrees with the notion introduced by Gaiotto and Johnson-Freyd.

Our main result is that higher idempotent-completion preserves the existence of adjoints at all categorical levels. Applying this completion to iterated deloopings of the monoidal unit therefore yields symmetric monoidal (∞,n)-categories in which every object is fully dualizable. For enrichment over vector spaces, we identify them with the Morita n-categories of separable multifusion (n−2)-categories, establishing their full dualizability. We also use this description to establish their higher semiadditivity.