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 symmetric monoidal categories (SMC‑categories) into categories enriched over permutative categories (PC‑categories). This is the “many‑0‑cells” analogue of the classical strictification results for symmetric monoidal and bimonoidal categories. The authors first recall that the classifying space of a symmetric monoidal category is an E∞‑space, whose group completion yields the zeroth space of a K‑theory spectrum. When a symmetric monoidal category carries a second (multiplicative) monoidal structure, the resulting bimonoidal category gives rise to a ring spectrum via K‑theory. However, in practice the multiplicative structure is often only weakly compatible with the additive one, leading to the notion of an SMC‑category: a bicategory whose hom‑categories are equipped with a symmetric monoidal structure, and whose horizontal composition is bilinear up to coherent isomorphisms.

The paper defines SMC‑categories precisely (Definition 2.1), listing the required distributivity, associativity, and unit isomorphisms, all of which must be monoidal natural transformations. A series of examples (rings, spans, bimodules, the 2‑category SymMon, and its sub‑2‑categories Perm, Permu, etc.) illustrate that many familiar constructions naturally form SMC‑categories.

The strict counterpart, a PC‑category (Definition 3.1), is a 2‑category where each hom‑category is a permutative category (strict symmetric monoidal), and the left and right composition functors are respectively strong symmetric monoidal (with strict units on the right). Only one of the two possible distributivity laws is required to be strict; the other is a coherent isomorphism, reflecting the impossibility of making both strict without also making commutativity strict.

The main result (Theorem 1.2) states that every SMC‑category is biequivalent, via an SMC‑functor, to a PC‑category. Two independent proofs are given.

Proof I (Section 5) follows the classical Isbell‑type strictification. For a given SMC‑category B, a new 2‑category B′ is built whose 1‑cells are formal strings of composable 1‑cells of B, with empty strings serving as strict units. Concatenation of strings provides strict horizontal composition. A surjective functor π collapses each string to the actual composite in B, and a canonical inclusion η embeds B as length‑one strings. One checks that π and η are inverse up to biequivalence, and that the monoidal structures on hom‑categories lift to strict permutative structures on B′. Thus B′ is a PC‑category biequivalent to B.

Proof II (Section 7) uses the Yoneda embedding. For any SMC‑category B, consider the 2‑functor B̂ = SMC‑Cat(–, B). By definition, each hom‑category B̂(a,b) is a category of SMC‑functors, which inherits a permutative structure from pointwise addition in B. The Yoneda embedding B → B̂ is fully faithful and essentially surjective on objects, yielding a biequivalence. Because B̂ is already a PC‑category, this furnishes the desired strictification without any explicit construction of strings.

Section 8 discusses how the same arguments apply to bimonoidal categories, especially when the multiplicative monoidal structure is symmetric, thereby recovering the known strictification of bimonoidal categories as a special case.

Consequently, any SMC‑category—such as the bicategory of spans, bimodules, or the 2‑category of symmetric monoidal categories—can be replaced, up to biequivalence, by a PC‑category. This replacement preserves the data needed for K‑theory constructions, ensuring that the resulting K‑theory spectrum inherits the expected ring or module structure. The paper thus provides both a concrete, hands‑on strictification procedure and an abstract, Yoneda‑based coherence argument, enriching the toolbox for higher‑categorical algebra and its applications in algebraic topology and K‑theory.