Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras

We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such notion was introduced by Hu and Kriz, but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.

💡 Research Summary

The paper addresses a fundamental problem in the theory of pseudo‑algebras: how to decide which coherence diagrams must be required to commute. Pseudo‑algebras arise when the underlying set‑based algebraic structure is categorified—operations become functors and equations become natural isomorphisms (coherence isomorphisms). In addition to these isomorphisms one must impose a collection of commutative diagrams, the “coherence diagrams”, to control the interaction of the isomorphisms.

Earlier work by Hu and Kriz (and earlier by Laplaza) suggested a very liberal scheme: any diagram that can be “reasonably expected” to commute should be imposed. In practice this means that whenever a word in the algebra can be transformed into another word by two different sequences of identities, the corresponding diagram of coherence isomorphisms must commute. The authors show that this prescription is far too strong. For example, in a commutative monoid the word a⊕a can be turned into itself either by the empty sequence of identities or by applying the commutativity identity a⊕b = b⊕a with b = a. The resulting diagram forces the coherence isomorphism τ_{aa} to be the identity, which would make a pseudo‑commutative monoid stricter than a symmetric monoidal category—an undesirable restriction (see Proposition 2.5).

To remedy this, the authors introduce Laplaza sets. A Laplaza set S for a theory T is a collection of words S(n) ⊆ T(n) (for each arity n) that specifies precisely which words are allowed to generate coherence diagrams. The construction proceeds as follows:

1. Theories and categorical theories – The paper recalls the notion of a Lawvere‑style theory T as a functor T : Γ → Sets equipped with composition operations γ satisfying associativity, unit, and two equivariance axioms. The internal version replaces Sets by Cat, yielding a categorical theory.

2. Operads – By restricting the morphisms of Γ to bijections and the equivariance axioms accordingly, the authors obtain “operads” that encode operations where each variable appears exactly once. Free theories on such operads are generated by operations and equations that do not repeat variables on either side.

3. Graphical pre‑theories and free categorical theories – Given a Laplaza set S, they first form a graphical pre‑theory G_S whose objects are the free theory on the collection {S(n)}. For each pair of objects a,b mapping to the same word in S(n) they insert a single arrow ι_{a,b}.

4. Quotient to obtain T′_S – Starting from the free categorical theory F_S on G_S, they impose two families of relations: (i) ι_{a,b} is identified with the inverse of ι_{b,a}, and (ii) any two arrows with the same source and target that project to the same word in S are identified. The resulting quotient is the categorical theory T′_S.

5. Pseudo‑algebras with respect to S – A pseudo‑algebra for (T,S) is simply a morphism of categorical theories T′_S → End_Cat(X). This morphism interprets the abstract isomorphisms of T′_S as concrete coherence isomorphisms in X, and forces exactly those diagrams whose source and target words lie in S to commute.

The Laplaza‑set framework thus selectively weakens the coherence requirements: only diagrams built from words in S are enforced, while “bad” diagrams (such as the τ_{aa} diagram above) are omitted.

The authors illustrate the utility of Laplaza sets with several examples:

• Commutative monoids – By taking S to consist of words a⊕b with a≠b, the problematic diagram for a⊕a is excluded, and pseudo‑commutative monoids become precisely symmetric monoidal categories.

• Commutative semirings (bimonoidal categories) – The distributivity law involves a word where a variable appears twice on one side. Laplaza’s original coherence conditions for distributive monoidal categories are recovered by an appropriate Laplaza set that permits this specific repetition while still forbidding other unwanted repetitions.

• World‑sheets (Riemann surfaces with parametrized boundaries) – These structures require dynamically indexed operations (gluing depends on the number of boundary components). They are modeled by 2‑theories. By defining 2‑operads and 2‑Laplaza sets, the authors show how to generate the correct set of coherence diagrams for pseudo‑world‑sheet algebras.

• Multi‑sorted algebras – Rings together with modules, or any I‑sorted algebra, fit into the same pattern: the free theory on the appropriate operad captures the operations, and a Laplaza set selects the admissible coherence diagrams.

The paper proves that the construction preserves the essential results of the earlier works (