Enriched $\infty$-categories via non-symmetric $\infty$-operads (1312.3178v4)

Abstract: We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This is analogous to the advantages of $\infty$-categories over more rigid models such as simplicial categories - for example, the resulting $\infty$-categories of functors between enriched $\infty$-categories automatically have the correct homotopy type. We construct the homotopy theory of $\mathcal{V}$-enriched $\infty$-categories as a certain full subcategory of the $\infty$-category of "many-object associative algebras" in $\mathcal{V}$. The latter are defined using a non-symmetric version of Lurie's $\infty$-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric $\infty$-operads. While we may regard these "many-object" algebras as enriched $\infty$-categories, we show that it is precisely the full subcategory of "complete" objects (in the sense of Rezk, i.e. those whose space of objects is equivalent to its space of equivalences) which are local with respect to the class of fully faithful and essentially surjective functors. Lastly, we present some applications of our theory, most notably the identification of associative algebras in $\mathcal{V}$ as a coreflective subcategory of pointed $\mathcal{V}$-enriched $\infty$-categories as well as a proof of a strong version of the Baez-Dolan stabilization hypothesis.

Enriched $\infty$-Categories via Nonsymmetric $\infty$-Operads

The paper "Enriched $\infty$-Categories via Nonsymmetric $\infty$-Operads" by David Gepner and Rune Haugseng establishes a comprehensive framework for enriching $\infty$-categories within an arbitrary monoidal $\infty$-category $\mathcal{V}$. This work contributes to the broader discussion on categorical algebra and homotopy theory by generalizing the classical notion of enriched category to the context of higher categories.

Overview

The central construct of the paper is the development of a theory of enriched $\infty$-categories that are homotopy-coherent, providing a natural generalization of the established notions in category theory while extending them to the $\infty$-categorical context. The research is particularly useful for instances where enriched $\infty$-categories emerge from model categories, such as differential graded categories (dg-categories), spectral categories, and $(\infty,n)$-categories.

The authors propose a formulation based on categorical operads, leveraging the nonsymmetric variant of Lurie's $\infty$-operads. A significant feature of their approach is the adoption of a non-symmetric framework, aligning well with the theory's applications and examples of interest, and enhancing the representation of homotopy-coherent structures.

Strong Numerical Results and Contradictory Claims

While the paper does not present numerical results in a conventional experimental sense, it advances strong theoretical claims, particularly postulating the equivalence of certain categories within different frameworks. Notable is the demonstration that $\mathcal{V}$ enriched $\infty$-categories can be identified as a coreflective subcategory of certain pointed $\mathcal{V}$-enriched $\infty$-categories, alongside proving a strong version of the Baez-Dolan stabilization hypothesis.

Implications and Future Developments

The implications of this research are manifold, extending both theoretical and practical dimensions of category theory. On the theoretical side, it set a clear precedent for representing complex mathematical concepts within an $\infty$-categorical framework. Practically, understanding $\mathcal{V}$-enriched $\infty$-categories promises advancements in algebraic topology, representation theory, and algebraic geometry, among other fields.

In terms of future research, the paper opens avenues for exploring deeper connections between $\infty$-categories and derived category structures. Another potential direction involves leveraging this enriched category theory within computational domains, offering conceptual frameworks for managing complex homotopy types and iterative processes.

In summary, the work of Gepner and Haugseng presents a compelling approach to enriching $\infty$-categories, ensuring homotopy-coherence, and establishing the groundwork for subsequent developments in higher category theory. The blend of robust theoretical foundations with practical applicability sets the stage for comprehensive explorations in mathematics and theoretical computer science.