Free rigid commutative algebras

Abstract: We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one.

• The paper establishes a universal construction of free rigid commutative algebras in weakly 2-presentable symmetric monoidal (∞,2)-categories via oplax colimits over a 1D framed cobordism category.
• It introduces novel adjointability and dualizability lemmas to ensure that the constructed algebra is rigid and fully embeds into a compactly generated counterpart.
• It provides explicit formulae for hom-objects and endomorphism algebras, enabling precise computations in enriched higher category theory and stable homotopy contexts.

Free Rigid Commutative Algebras in $(\infty,2)$-Categories

Introduction and Motivation

The construction and universal characterization of symmetric monoidal categories with duals has foundational importance across several mathematical domains, including tensor-triangular geometry and the theory of motives. The universal property for such categories has been classically exemplified by the cobordism hypothesis, which identifies the free symmetric monoidal $(\infty,1)$-category with duals on a generator with the $1$-dimensional framed cobordism category. In the setting of higher categories and enriched categorical frameworks, a unified and structurally transparent description of free rigid commutative algebras becomes necessary, particularly within presentably symmetric monoidal $(\infty,2)$-categories and their module categories.

The primary focus of the paper "Free rigid commutative algebras" (2604.01854) is to establish a general framework for constructing free rigid commutative algebras in weakly $2$-presentably symmetric monoidal $(\infty,2)$-categories, connecting the resulting objects to oplax colimits over the $1$-dimensional framed cobordism category. This approach both generalizes and refines previous constructions—encompassing enriched settings—and provides new avenues for explicit computation and further categorical rigidity results.

Main Contributions

Construction via Oplax Colimits over Cobordism

The core achievement is the identification of the free rigid commutative algebra generated by a dualizable object $b$ in a weakly $2$-presentably symmetric monoidal $(\infty,2)$-category $(\infty,1)$0 as an explicit oplax colimit over the $(\infty,1)$1-dimensional framed cobordism category $(\infty,1)$2: $(\infty,1)$3 where the symmetric monoidal functor $(\infty,1)$4 sends the positively oriented generator of $(\infty,1)$5 to $(\infty,1)$6.

This description is robust—applying not only in the context of $(\infty,1)$7 (presentable $(\infty,1)$8-categories) but also to $(\infty,1)$9 for any presentably symmetric monoidal $1$0-category $1$1, thereby capturing both ordinary and $1$2-enriched contexts.

Rigidification and Dualizability

The framework is grounded in the modern definition of a rigid commutative algebra object: a commutative algebra $1$3 in $1$4 for which both the unit and multiplication maps admit right adjoints satisfying strict module compatibility. Correspondingly, the underlying object of a rigid commutative algebra is dualizable in $1$5.

The categorical techniques developed ensure that the resulting oplax colimit not only carries the structure of a commutative algebra, but is rigid, benefiting from novel adjointability and dualizability lemmas developed using $1$6-categorical methods.

Explicit Formulae and Calculational Tools

The paper derives explicit formulae for hom-objects in the free rigid commutative algebra. Critically, the endomorphism object of the unit in $1$7 is the free commutative algebra on the homotopy $1$8-orbits of the dimension object of $1$9, i.e.: $(\infty,2)$0 This elegantly generalizes classical results (e.g., those of Deligne–Milne for additive categories) and gives transparent computational access to the morphisms between tensor powers, pairing them via combinatorial data of the cobordism category.

The explicit module structure for internal homs and mapping objects is also unpacked, with coproducts indexed over bijections, and an explicit dependence on the structure dictated by the symmetric group actions and duality morphisms.

Embedding and Rigidification Results

A central byproduct is a new proof (and generalization) of the fact that any rigid commutative algebra in $(\infty,2)$1 embeds fully faithfully into a compactly-generated rigid commutative algebra, answering an open question in the field in the affirmative and without recourse to classical compactification arguments.

Further, the paper proves broad rigidification results: any locally rigid commutative algebra admits a fully faithful embedding into a rigid commutative algebra via an explicit adjoint, and this process is compatible (i.e., adjointable) under sufficiently nice $(\infty,2)$2-functors.

Implications and Theoretical Significance

The bridging of the ($(\infty,2)$3)-categorical perspective with enriched and presentably symmetric monoidal settings opens new computational and theoretical directions. In practical terms, this framework allows for direct construction and manipulation of free rigid categories (and their algebras) in settings relevant to enriched higher category theory, derived algebraic geometry, and stable homotopy theory.

The approach transcends the limitations of prior one-dimensional or purely $(\infty,2)$4-categorical results by allowing direct interface with dualizable and rigid objects not arising from presheaf categories (e.g., categories of sheaves of spectra on non-compact spaces), as well as handling enrichment without requiring underlying "atomic" generators.

Numerical and Structural Results

The structural description of internal endomorphism algebras as free commutative algebras on $(\infty,2)$5-homotopy orbits of dimension objects is a particularly strong result, clarifying the essential role of group actions in the "higher" categorical situation and enabling precise calculations well beyond the additive or classical context.

The assertion that every rigid commutative algebra embeds into a compactly generated one is affirmative and unexpected in general, broadening the foundational landscape of tensor-triangular and a stable category theory.

Connections to Existing and Future Work

This work generalizes results obtained by Barkan and Steinebrunner, employing a different, arguably more conceptual and morphism-oriented technique, and yielding transparent structural formulae. The innovation here lies in the method of construction (via oplax colimits) and the operability in broad categorical contexts.

The paper also relates in a nuanced way to work by Neuhauser on universal rigid commutative algebras, but improves on it by effectively parametrizing the base symmetric monoidal $(\infty,2)$6-category and providing universal free constructions within arbitrary such categories equipped with dualizable objects.

Future developments will likely focus on extending these constructions to $(\infty,2)$7-monoidal settings, investigating connections to higher-dimensional cobordisms, and further integrating these results with enriched and motivic homotopy theory, particularly in contexts where motivic equivalences or compact generation play critical roles.

Conclusion

This paper presents a rigorous, detailed solution to the problem of constructing free rigid commutative algebras in highly structured $(\infty,2)$8-categories, demonstrating both the theoretical depth and computational utility of an approach via oplax colimits over the framed $(\infty,2)$9-dimensional cobordism category. By consolidating and generalizing prior results, establishing explicit compositional and algebraic formulae, and achieving new embedding and rigidification theorems, the work stands as a substantial contribution to the theory of higher categorical algebra and its applications to stable homotopy and beyond.