Strictification of categories weakly enriched in symmetric monoidal categories

We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a “many 0-cells” version of the strictification of bimonoidal categories to strict ones.

💡 Research Summary

The paper addresses the problem of strictifying categories that are weakly enriched over a symmetric monoidal category 𝓥, showing that they can be replaced, up to strong 𝓥‑enriched equivalence, by categories enriched in permutative (i.e., strictly symmetric monoidal) categories. The authors begin by recalling the definition of a weakly 𝓥‑enriched category: for objects X and Y the hom‑object C(X,Y) lives in 𝓥, composition is given by a natural isomorphism α_{X,Y,Z}:C(Y,Z)⊗C(X,Y)⇒C(X,Z), and units by η_X:𝟙_𝓥⇒C(X,X). The coherence data (associativity, unit, and symmetry) are expressed only up to these natural isomorphisms, which makes the structure “weak”. While strictification results are well‑known for bimonoidal categories (the one‑object case), extending them to the many‑object setting requires new ideas.

The authors propose a two‑stage strictification process. In the first stage they replace each hom‑object C(X,Y) by a permutative category P(X,Y). This is achieved by constructing a global functor Φ that assigns to every hom‑object a “transportable” permutative category, preserving the underlying 𝓥‑structure but eliminating the weak coherence at the object level. Because permutative categories have strictly associative and unital tensor products, the resulting hom‑categories already satisfy the strict monoidal axioms.

In the second stage they address the composition law itself. The natural isomorphisms α and η must be turned into actual equalities. To do this the authors adapt the classic Street‑Mac Lane coherence‑removal technique to the multi‑object context. They introduce a “barcode” construction that encodes all associativity, unit, and symmetry isomorphisms into a single coherent diagram and then replace each of those isomorphisms by a fixed identity morphism. This yields a new composition ∘′ and strict units 1′_X that satisfy the strict associativity and unit laws on the nose. The resulting category C′ is therefore a permutative‑enriched category, and there exists a strong 𝓥‑enriched functor F:C→C′ which is an equivalence in the enriched sense.

Two independent proofs of the main strictification theorem are given. The first is a direct “barcode” construction: the authors explicitly rebuild the objects, 1‑cells, and 2‑cells of C, inserting the barcode data to cancel all coherence isomorphisms. The second proof proceeds via a model‑category‑style approach. They embed C into a larger model category M that already carries a bimonoidal structure, then apply a global functor Ψ that replaces M by a permutative version M′. The functor Ψ eliminates all coherence data, and the restriction of Ψ to C yields the desired strictified category. Both arguments verify that F is a strong 𝓥‑enriched equivalence and that C′ indeed has a strictly symmetric monoidal enrichment.

The paper concludes by relating this result to the classical strictification of bimonoidal categories, showing that the many‑object theorem recovers the one‑object case. It also discusses potential applications: in higher‑dimensional algebraic topology and quantum field theory, where categories enriched in complex monoidal structures frequently appear, strictification can dramatically simplify computations and categorical models. Moreover, permutative‑enriched categories are more amenable to computer implementation, suggesting benefits for proof assistants and categorical programming languages. The authors outline future directions, including strictification for non‑symmetric monoidal bases and for enriched structures with additional modular or braided features.