Coherence for Monoidal Endofunctors

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite products or coproducts. Monoidal endofunctors are interesting because they stand behind monoidal monads and comonads, for which coherence will be proved in a sequel to this paper.

💡 Research Summary

The paper addresses a fundamental coherence problem for monoidal endofunctors—functors (F:\mathcal{C}\to\mathcal{C}) on a monoidal category that preserve the tensor product and unit only up to a natural transformation, not necessarily an isomorphism. Traditional coherence results for monoidal functors and monads assume that the structural maps (\phi_{A,B}:FA\otimes FB\to F(A\otimes B)) and (\phi_0:I\to FI) are invertible. By relaxing this requirement, the authors open the door to a much broader class of constructions, including many monoidal monads and comonads whose unit or multiplication are not invertible (e.g., the list monad, probability distribution monad, or various comonadic stream structures).

The central technical tool is a “relational graph” semantics. Objects of the free monoidal endofunctor category are formal tensor expressions; morphisms are generated from the structural natural transformations (\phi), the identity, and, when present, symmetry, diagonal, or codiagonal maps. Each morphism is interpreted as a directed graph: vertices correspond to objects, edges to the basic natural transformations, and composition to graph concatenation. The authors define a functor (G) from the syntactic free category to the category of relational graphs (\mathbf{Rel}). Coherence then amounts to showing that (G) is faithful (different syntactic morphisms yield different graphs) and essentially surjective onto the intended class of graphs.

The paper proceeds in several stages. First, in the absence of symmetry, a free monoidal endofunctor category (\mathsf{Free}(F)) is built, and a normal‑form reduction system is introduced. The authors prove that every morphism reduces uniquely to a normal form, and that two morphisms have the same relational graph iff their normal forms coincide. This yields Theorem 1: (G:\mathsf{Free}(F)\to\mathbf{Rel}) is fully faithful, establishing coherence for non‑symmetric monoidal endofunctors.

Next, symmetry is added. The free symmetric monoidal endofunctor category (\mathsf{Free}{\mathrm{sym}}(F)) incorporates the braiding (\sigma{A,B}). The relational graph functor is extended by allowing “switch” nodes that represent the symmetry. Theorem 2 shows that even with these switches, the reduction system still yields unique normal forms, and the graph functor remains fully faithful. The proof carefully analyses the interaction between switches and the structural (\phi) maps, ensuring that all critical pairs are confluent.

The authors then consider additional natural transformations: a diagonal (\Delta_A:A\to A\otimes A) and a codiagonal (\nabla_A:A\otimes A\to A). These give rise to duplication and erasure nodes in the graphs, mirroring the behavior of copying and discarding in cartesian or cocartesian monoidal categories. Separate coherence theorems (Theorems 3 and 4) are proved for each case, again using a normal‑form strategy that now includes duplication/erasure reduction rules. The confluence arguments are extended to cover interactions between duplication, erasure, and the original (\phi) maps.

Finally, the paper specializes to monoidal structures given by finite products ((\times,1)) or coproducts ((+,0)). In these settings, diagonals and codiagonals are automatically present, so the previous results apply directly. The authors illustrate the theory with concrete examples from Set (product and coproduct) and from programming language semantics (e.g., the list monad as a monoidal monad on the cartesian category of sets).

Throughout, the relational graph perspective provides a uniform visual language that captures the essence of coherence without relying on invertibility. The results not only generalize classic Mac Lane coherence for monoidal functors but also lay the groundwork for a sequel in which the authors will prove coherence for monoidal monads and comonads themselves. Potential applications include effect systems in programming languages, resource‑sensitive type theories, and categorical models of quantum computation, where non‑invertible structural maps are the norm rather than the exception.