Strictification, Comparison, and Foundational Limits

Weak categorical composition is often strictifiable: every bicategory is biequivalent to a strict two-category. The theorem preserves objects only up to essential surjectivity and each hom-category up to equivalence; it does not turn the original associator into an equality inside the original model. More importantly for QFT, strictifying an underlying bicategory does not automatically preserve topology, smooth dependence, adjoints, positivity, unbounded-operator domains, or observable functors. Those structures must be included in the comparison.

Required background. Factorization Comparison Theorems and Their Limits supplies a model for conditional equivalence theorems. Higher Morita Categories and Theories as Objects supplies the bicategories of phases and interfaces. Homotopical Renormalization and Effective-Theory Maps supplies quasi-isomorphisms that preserve formal deformation problems. Helpful background. Chain Homotopy, Quasi-Isomorphisms, and Derived Vocabulary distinguishes equality from homotopy equivalence. Duality Claims, Dictionaries, Regimes, and Evidence supplies comparison discipline. Topological Twists and the Geometric–Langlands Interface gives a setting where categorical equivalence must not be confused with equivalence of untwisted theories.

Coherence for bicategories

A bicategory $\mathcal B$ has objects, categories $\mathcal B(A,B)$ of one- and two-morphisms, composition functors, identity one-morphisms, and invertible associator and unitor two-morphisms. For four composable one-morphisms, the associators obey the pentagon identity. A strict two-category is the special case in which associativity and unit laws are literal equalities.

A weak functor $F:\mathcal B\to\mathcal C$ is a biequivalence when

1. each $F_{A,B}:\mathcal B(A,B)\to\mathcal C(FA,FB)$ is an equivalence of categories; and
2. every object of $\mathcal C$ is equivalent to some $FA$.

These are local equivalence and essential surjectivity, respectively. Leinster’s coherence theorem states that every bicategory is biequivalent to some strict two-category 2004, Theorem 1.5.15, p. 32. His proof embeds the bicategory by a bicategorical Yoneda functor into a strict functor two-category and takes the full strict sub-two-category on representables. A different concrete model uses strings of composable one-morphisms: concatenation of strings is strictly associative, while evaluation composes the string in the original bicategory using its associators.

The theorem’s conclusion is exactly biequivalence. It does not give an isomorphism of bicategories, and it does not say the strict model is unique. Transporting extra structure requires an enriched or structured version of the theorem and coherence of the transported data.

First application: a strict model for defect fusion

Return to Fusion, Junctions, and Endpoints. In the Morita bicategory, objects are algebras representing topological phases, one-morphisms are bimodules representing interfaces, and two-morphisms are bimodule maps representing point junctions. For composable interfaces

$${}_AM_B,\qquad {}_BN_C,\qquad {}_CP_D,$$

the two fusion orders

$$(M\otimes_BN)\otimes_CP \quad\text{and}\quad M\otimes_B(N\otimes_CP)$$

are connected by a canonical bimodule isomorphism. This associator is physical data at the junction level; its pentagon ensures that fourfold fusion is coherent.

In a string strictification, a one-morphism $A\to D$ can be represented by the word $(M,N,P)$. Composition is concatenation, so $(M,N)P=M(N,P)$ as words. The evaluation weak functor sends the word to a chosen parenthesized relative tensor product and sends the strict equality of concatenations to the original associator. Junction intertwiners are transported through the equivalences of hom-categories. Thus objects, interfaces, junctions, and the coherent associator survive up to the specified biequivalence.

This is consistent with the algebraic target used for extended two-dimensional field theories: $\operatorname{Alg}_2$ has algebras, bimodules, and intertwiners, and fully extended oriented theories select separable symmetric Frobenius objects Schommer-Pries 2011, §3.8, pp. 230–244. Strictification can simplify the compositional bookkeeping inside that bicategory. It does not prove that a proposed continuum defect theory lands in $\operatorname{Alg}_2$, nor that its analytic fusion is the algebraic relative tensor product.

An independent check is the pentagon. Evaluate the five strict paths between parenthesizations of a word of four bimodules. Because concatenation is literally associative, the boundary of the pentagon is the identity in the strict model. Under evaluation it becomes the standard pentagon composite of Morita associators, which is also the identity. If the proposed evaluation fails this check, it is not a weak two-functor.

What strictification does not preserve automatically

Suppose the original QFT interfaces are Hilbert bimodules with topologies, dense domains for unbounded junction operators, involutions, and positivity conditions. Forget these data, strictify the underlying algebraic bicategory, and then declare the result an equivalent physical theory. This is the adversarial failure. Biequivalence of the forgotten algebraic layer contains no map of topological vector spaces, no control of closures or operator domains, and no assertion that positive inner products or adjoints are preserved.

The same problem occurs with observables. A QFT comparison needs a specified observable functor—perhaps factorization algebras on opens, state spaces on boundaries, or correlators—and a natural equivalence intertwining it. Biequivalence of phase/interface categories alone does not force those functors to agree. Nor does strictification of a bicategory imply that all higher categories can be made fully strict without loss; already at the next categorical level the appropriate semistrict models retain nontrivial interchange and coherence.

There are therefore three separate conclusions:

• categorical coherence: the bicategory admits a biequivalent strict two-category;
• structured coherence: selected monoidal, pivotal, topological, or $*$-data are transported by a theorem that names them;
• physical equivalence: observables, states, locality, and analytic domains are intertwined in the claimed regime.

Only the first follows from the general coherence theorem. The second and third need additional hypotheses and comparison maps. Conversely, observing the same fusion ring or the same decategorified dimensions does not reconstruct the associators, junction categories, or analytic data.

Exercises

Explain why local equivalence without essential surjectivity is not a biequivalence.

Solution

Take the inclusion of the full one-object subcategory into a discrete category with two inequivalent objects. It is an equivalence on the single represented hom-category but misses the second object up to equivalence. The same construction viewed as locally discrete bicategories is locally an equivalence and not essentially surjective, hence not a biequivalence.

What extra check is needed to claim that strictification preserves adjoints of interfaces?

Solution

One must transport the chosen adjoint one-morphism together with evaluation and coevaluation two-morphisms and verify the triangle identities after transport. An unstructured biequivalence gives equivalences of hom-categories, but the claim about specified adjoint data requires showing that these particular morphisms and identities are carried to their counterparts.

References

• Leinster 2004, Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series 298, Cambridge University Press. Open PDF
• Schommer-Pries 2011, The Classification of Two-Dimensional Extended Topological Field Theories, PhD thesis, University of California, Berkeley. Open PDF