Categorical Weisfeiler-Lehman Framework

• The categorical Weisfeiler-Lehman framework is a family of methods that redefines WL refinement through functorial lifting onto graded posets and combinatorial complexes.
• It separates the choice of lifting functor from neighborhood aggregation, providing a principled invariant for graph and hypergraph isomorphism tests.
• Empirical results demonstrate high expressivity and improved performance on hypergraph classification and topological benchmarks.

Searching arXiv for the specified papers to ground the article in the current literature. The categorical Weisfeiler-Lehman framework denotes a family of formulations that recast WL-style refinement through explicitly categorical structure. In the graded-poset formulation, lifting is formalized as a functor from a data category to the category of graded posets, after which a universal graded WL procedure is applied (Choi et al., 6 Feb 2026). In a closely related topological formulation, WL refinement on combinatorial complexes is treated as a functorial, isomorphism-invariant process on a category of combinatorial complexes, yielding the Combinatorial Complex Weisfeiler-Lehman framework (Chen et al., 1 May 2026). A historically distinct but relevant precursor formulates graph isomorphism as an orbit problem and introduces approximate categories $\mathcal{C}_d(V)$ whose isomorphisms refine orbital indistinguishability and strictly extend fixed-dimensional WL on certain instances (Derksen, 2010). Taken together, these works place WL, lifting, higher-order message passing, and isomorphism testing within a categorical vocabulary.

1. Scope and historical formulations

The literature uses the categorical perspective in at least three closely related senses. “Weisfeiler and Lehman Go Categorical” introduces the Categorical Weisfeiler-Lehman (CatWL) framework, in which a functor $F:\mathcal{C}\to\mathbf{Poset}$ lifts objects from a source data category to graded posets, and WL refinement is then performed on the lifted poset. “Weisfeiler Lehman Test on Combinatorial Complexes” develops the Combinatorial Complex Weisfeiler-Lehman (CCWL) test as an axiomatic-style extension of WL to combinatorial complexes and explicitly casts the construction as functorial refinement on a category $\mathrm{CombCx}$. “The Graph Isomorphism Problem and approximate categories” instead constructs approximate categories $\mathcal{C}_d(V)$ from truncated neighborhood algebras and uses categorical isomorphism as an orbit certificate for graph isomorphism [(Choi et al., 6 Feb 2026); (Chen et al., 1 May 2026); (Derksen, 2010)].

These formulations share a common structural move: WL is no longer treated as a graph-specific heuristic, but as a refinement procedure induced by categorical data such as morphisms, functors, graded order, or truncated algebraic neighborhoods. The frameworks differ, however, in their target objects and technical aims. CatWL is centered on graded posets as a universal lifting domain. CCWL is centered on combinatorial complexes and topological message passing. The approximate-category construction is centered on algebraic group actions and orbit testing.

Formulation	Target objects	Core statement
CatWL	Graded posets	Lifting is a functor $F:\mathcal{C}\to\mathbf{Poset}$; GWL is applied to $F(X)$
CCWL	Combinatorial complexes	CCWL extends WL to four intrinsic neighborhood relations on cells
Approximate categories	$\mathcal{C}_d(V)$ from truncated neighborhood algebras	WL reduces to AC, but AC does not reduce to WL

A common misconception is that “categorical” names a single algorithm. The cited papers instead show a family of categorical reorganizations of WL: functorial lifting to graded posets, functorial refinement on combinatorial complexes, and approximate categorical isomorphism for orbit problems. This suggests that the categorical aspect is best understood as a design principle for defining invariants and message-passing topologies, rather than as one fixed refinement rule.

2. Graded posets as a universal WL domain

CatWL identifies graded posets as a universal target domain for lifting diverse data structures, including graphs, hypergraphs, simplicial complexes, and regular cell complexes. A graded poset $P$ is a poset $(P,<)$ equipped with a dimension function $\dim:P\to\mathbb{N}$ such that $F:\mathcal{C}\to\mathbf{Poset}$0 implies $F:\mathcal{C}\to\mathbf{Poset}$1, and if there is no $F:\mathcal{C}\to\mathbf{Poset}$2 with $F:\mathcal{C}\to\mathbf{Poset}$3, the cover relation is denoted $F:\mathcal{C}\to\mathbf{Poset}$4 and satisfies $F:\mathcal{C}\to\mathbf{Poset}$5 (Choi et al., 6 Feb 2026).

On such a poset, the Graded Weisfeiler-Lehman (GWL) test performs color refinement using four adjacency types induced by the Hasse relation. Starting from a constant coloring $F:\mathcal{C}\to\mathbf{Poset}$6, the update is

$F:\mathcal{C}\to\mathbf{Poset}$7

where $F:\mathcal{C}\to\mathbf{Poset}$8 are boundary adjacencies, $F:\mathcal{C}\to\mathbf{Poset}$9 are co-boundary adjacencies, $\mathrm{CombCx}$0 are lower adjacencies via a shared boundary face, and $\mathrm{CombCx}$1 are upper adjacencies via a shared co-boundary coface. Two graded posets are isomorphic by GWL if the coloring histograms are equal for all $\mathrm{CombCx}$2 (Choi et al., 6 Feb 2026).

The categorical step is then straightforward. Given a functor $\mathrm{CombCx}$3, the $\mathrm{CombCx}$4-Categorical WL test lifts $\mathrm{CombCx}$5 to $\mathrm{CombCx}$6 and applies GWL there. Because functorial lifting preserves isomorphisms, $\mathrm{CombCx}$7-CatWL provides a necessary condition for isomorphism in the source category. The paper also defines a neural relaxation, CatMPN, whose neighborhoods are exactly those induced by the Hasse diagram of $\mathrm{CombCx}$8 and whose expressivity matches CatWL when aggregation and update functions are injective and the network has sufficient depths and widths (Choi et al., 6 Feb 2026).

This graded-poset formalization is significant because it separates two design choices that are often conflated in higher-order learning: the choice of lifting functor and the choice of refinement rule. The message-passing topology is strictly determined by the choice of functor. Consequently, the categorical framework does not prescribe a unique architecture; it prescribes a principled mechanism for deriving one.

3. Functorial lifting, hypergraphs, and induced architectures

For hypergraphs, CatWL develops two explicit functors from the category $\mathrm{CombCx}$9 to $\mathcal{C}_d(V)$0. In $\mathcal{C}_d(V)$1, an object is a triple $\mathcal{C}_d(V)$2 with $\mathcal{C}_d(V)$3, and a morphism $\mathcal{C}_d(V)$4 consists of maps on vertices and hyperedges satisfying $\mathcal{C}_d(V)$5. The first lifting is the incidence functor $\mathcal{C}_d(V)$6, which maps $\mathcal{C}_d(V)$7 to the graded poset $\mathcal{C}_d(V)$8 with $\mathcal{C}_d(V)$9, $F:\mathcal{C}\to\mathbf{Poset}$0, and $F:\mathcal{C}\to\mathbf{Poset}$1 iff $F:\mathcal{C}\to\mathbf{Poset}$2. The second is the symmetric simplicial complex functor $F:\mathcal{C}\to\mathbf{Poset}$3, which constructs simplices internal to each hyperedge and orders them by inclusion within that hyperedge, thereby preserving the full internal subset geometry of each hyperedge (Choi et al., 6 Feb 2026).

These lifts induce distinct message-passing topologies. The incidence poset yields a two-level structure enabling node-hyperedge interactions, but the categorical derivation also enforces aggregation over lower and upper adjacencies, capturing intersection geometries between hyperedges and between nodes via shared hyperedges. The symmetric simplicial construction stratifies each hyperedge by cardinality and produces hierarchical interactions between faces and cofaces, allowing explicit modeling of shared-face geometry within large hyperedges (Choi et al., 6 Feb 2026).

The corresponding neural architectures are hypergraph isomorphism networks instantiated from the generic CatMPN template. For features $F:\mathcal{C}\to\mathbf{Poset}$4, messages are aggregated over $F:\mathcal{C}\to\mathbf{Poset}$5, $F:\mathcal{C}\to\mathbf{Poset}$6, $F:\mathcal{C}\to\mathbf{Poset}$7, and $F:\mathcal{C}\to\mathbf{Poset}$8, and a learnable update combines them. Theorem 9 states that $F:\mathcal{C}\to\mathbf{Poset}$9-CatWL is at least as powerful as the $F(X)$0-CatMPN, and that equality of expressive power holds under injective update and aggregation functions with sufficient capacity. The framework further proves that both $F(X)$1-CatWL and $F(X)$2-CatWL are not less powerful than HWL; the proofs use explicit counterexamples where HWL fails to distinguish two non-isomorphic hypergraphs but the categorical refinements succeed at $F(X)$3 (Choi et al., 6 Feb 2026).

The paper also establishes adjacency simplification results. Under the boundary-size condition $F(X)$4 for any $F(X)$5 with $F(X)$6, the full four-adjacency update is equivalent in expressive power to restricted variants: boundary, lower, and upper adjacencies suffice, and even boundary with upper adjacencies suffices. This is an important correction to the intuition that more adjacency types are automatically more informative; under the stated condition, some of them are theoretically redundant (Choi et al., 6 Feb 2026).

Empirically, the induced architectures I-HIN and S-HIN achieve best performance on 5 of 6 hypergraph classification benchmarks. I-HIN dominates on dense graphs with many hyperedges per graph, while S-HIN performs best on sparse graphs with large hyperedge cardinalities, where internal sub-relations within hyperedges carry discriminative information. This division of labor reinforces the central categorical claim: the lift, rather than the downstream update rule alone, determines which structural information becomes visible to refinement (Choi et al., 6 Feb 2026).

4. Combinatorial complexes and the CCWL generalization

CCWL extends the WL paradigm from graphs and domain-specific topological structures to combinatorial complexes. A combinatorial complex is a triple $F(X)$7 consisting of a set $F(X)$8, a subset $F(X)$9, and an order-preserving rank function $\mathcal{C}_d(V)$0 such that for all $\mathcal{C}_d(V)$1, $\mathcal{C}_d(V)$2 and $\mathcal{C}_d(V)$3. A cell $\mathcal{C}_d(V)$4 with $\mathcal{C}_d(V)$5 is a $\mathcal{C}_d(V)$6-cell, and the dimension is $\mathcal{C}_d(V)$7. The incidence structure is given by the boundary relation $\mathcal{C}_d(V)$8, with strict boundary relation $\mathcal{C}_d(V)$9 when there is no intermediate cell (Chen et al., 1 May 2026).

CCWL formalizes four neighborhood relations on cells. For a cell $P$0, the boundary adjacent set is $P$1, the co-boundary adjacent set is $P$2, the lower adjacent set is $P$3 for same-rank cells sharing a $P$4-cell, and the upper adjacent set is $P$5 for same-rank cells sharing a $P$6-cell. Shared-bridge sets $P$7 and $P$8 are intersections of boundary and co-boundary neighborhoods, respectively (Chen et al., 1 May 2026).

The CCWL update refines colors rank-wise. With coloring $P$9 and an injective multiset hashing function, the update is

$(P,<)$0

where $(P,<)$1 and $(P,<)$2 record colors on boundary and co-boundary cells, and $(P,<)$3 and $(P,<)$4 record same-rank neighbors together with shared bridge colors. Initialization is rank-wise, and refinement stops at stabilization. Lemma 4.4 gives isomorphism invariance: if $(P,<)$5, then the color multisets agree at every iteration (Chen et al., 1 May 2026).

The framework’s main expressive result is Theorem 4.8: CCWL with

$(P,<)$6

is as powerful as the full generalized update rule using all four adjacency types. The proof sketch recovers explicit boundary and co-boundary multisets from lower and upper adjacency information by projection onto bridge components and bridge multiplicity partitioning. Hence, upper and lower adjacencies suffice. The paper further states that CCWL is at least as powerful as 1-WL, simulates graph 1-WL, hypergraph WL, simplicial WL, and cellular WL via appropriate liftings, and is strictly more expressive than 1-WL, HWL, SWL, and CWL under faithful liftings (Chen et al., 1 May 2026).

This makes combinatorial complexes a unifying topological domain analogous, in spirit, to graded posets in CatWL. The difference is that CCWL’s unification is phrased intrinsically in terms of cell incidence and same-rank adjacencies inside a combinatorial complex, whereas CatWL emphasizes the external choice of a functor into a universal target category.

5. Categorical structure, invariance, and neural realization

CCWL gives an explicit categorical-style perspective through the category $(P,<)$7. Its objects are triples $(P,<)$8 as above, and its morphisms are structure-preserving maps that preserve rank and the four adjacency relations $(P,<)$9, $\dim:P\to\mathbb{N}$0, $\dim:P\to\mathbb{N}$1, and $\dim:P\to\mathbb{N}$2. Isomorphisms are bijective morphisms. A labeling functor $\dim:P\to\mathbb{N}$3 maps each combinatorial complex to the sequence of CCWL colorings $\dim:P\to\mathbb{N}$4, while a morphism acts by reindexing colors along the structure-preserving map. Lemma 4.4 establishes isomorphism invariance of these outputs, and HASH injectivity ensures that label refinement is a natural transformation with respect to incidence-preserving maps. The paper also notes a monoidal perspective: disjoint union acts like a coproduct, and because CCWL and CCIN use permutation-invariant multiset aggregation and readout, this invariance is preserved under disjoint unions up to aggregation of color multisets (Chen et al., 1 May 2026).

The neural instantiation is the Combinatorial Complex Isomorphism Network (CCIN). A general CCNN layer aggregates messages over the four neighborhoods $\dim:P\to\mathbb{N}$5, with neighborhood-specific message maps and a permutation-invariant aggregator. Using the sufficiency theorem, the update can be reduced to

$\dim:P\to\mathbb{N}$6

and the rank-adaptive CCIN layer is

$\dim:P\to\mathbb{N}$7

Lemma 5.1 states that CCIN is as powerful as CCWL when using injective neighborhood aggregators and sufficient MLP layers (Chen et al., 1 May 2026).

CatWL provides a parallel neural statement. CatMPN updates features by messages over boundary, co-boundary, lower, and upper adjacencies on the lifted poset $\dim:P\to\mathbb{N}$8. Theorem 9 gives the usual discrete-to-neural equivalence: CatWL is at least as powerful as CatMPN, and equality follows under injective aggregation and update functions with sufficient width and depth. In both frameworks, categorical invariance is therefore not merely descriptive; it becomes an expressivity theorem for specific message-passing architectures (Choi et al., 6 Feb 2026).

A historically different categorical realization appears in approximate categories $\dim:P\to\mathbb{N}$9. There the objects are elements of $F:\mathcal{C}\to\mathbf{Poset}$00 or affine subspaces of $F:\mathcal{C}\to\mathbf{Poset}$01, morphisms are truncated algebraic neighborhoods defined through

$F:\mathcal{C}\to\mathbf{Poset}$02

and composition is induced by comultiplication on the coordinate ring of the acting algebraic group. The construction yields isomorphism tests AC$F:\mathcal{C}\to\mathbf{Poset}$03 that run in polynomial time for fixed $F:\mathcal{C}\to\mathbf{Poset}$04 over suitable finite fields and strictly extend WL on Cai-Fürer-Immerman instances (Derksen, 2010). This is not a message-passing formulation, but it shares the same broad aim: replacing graph-local refinement by a categorical structure whose morphisms encode higher-order equivalence.

6. Expressivity, empirical profile, and open directions

Across these formulations, expressivity results consistently state that the categorical versions subsume domain-specific WL baselines. For hypergraphs, both $F:\mathcal{C}\to\mathbf{Poset}$05-CatWL and $F:\mathcal{C}\to\mathbf{Poset}$06-CatWL are not less powerful than HWL, and CatMPN matches CatWL under injectivity assumptions (Choi et al., 6 Feb 2026). For combinatorial complexes, CCWL is at least as powerful as 1-WL, simulates graph 1-WL, hypergraph WL, simplicial WL, and cellular WL through liftings, and is strictly more expressive than those variants under faithful liftings (Chen et al., 1 May 2026). For approximate categories, WL reduces to AC through constructible equivariants and functors, but AC does not reduce to WL; over $F:\mathcal{C}\to\mathbf{Poset}$07, AC$F:\mathcal{C}\to\mathbf{Poset}$08 distinguishes Cai-Fürer-Immerman pairs that no fixed-dimensional WL can distinguish (Derksen, 2010).

The empirical profile is similarly consistent with the theoretical claims. On hypergraph classification, I-HIN and S-HIN achieve best performance on 5 of 6 datasets, with I-HIN favored in dense regimes and S-HIN favored when internal hyperedge geometry matters (Choi et al., 6 Feb 2026). On topological benchmarks, CCIN attains 96.4% on MUTAG, 78.3% on IMDB-B, 93.4% on REDDIT-B, ROC-AUC $F:\mathcal{C}\to\mathbf{Poset}$09 on MOLHIV, AP $F:\mathcal{C}\to\mathbf{Poset}$10 on PEPTIDES-FUNC, MAE $F:\mathcal{C}\to\mathbf{Poset}$11 on PEPTIDES-STRUCT, MAE $F:\mathcal{C}\to\mathbf{Poset}$12 and $F:\mathcal{C}\to\mathbf{Poset}$13 on ZINC-small without and with edge features, and MAE $F:\mathcal{C}\to\mathbf{Poset}$14 on ZINC-FULL (250K). The ablations are notable: using only $F:\mathcal{C}\to\mathbf{Poset}$15 and $F:\mathcal{C}\to\mathbf{Poset}$16 improves over the full four-neighborhood model on several datasets, including RDT-B, RDT-M, NCI1, and NCI109, supporting the practical value of the upper/lower sufficiency theorem (Chen et al., 1 May 2026).

The main assumptions are explicit. CatWL requires a functor $F:\mathcal{C}\to\mathbf{Poset}$17 that preserves composition and identity and targets graded posets with well-defined cover relations and dimensions; equivalence with CatMPN requires injective aggregation and update maps (Choi et al., 6 Feb 2026). CCWL assumes rank-adjacent incidence, no free faces unless otherwise stated, injective HASH and permutation invariance, and uses a sufficiency proof that reconstructs boundary and co-boundary information from lower and upper adjacency under local incidence and bridge multiplicity partitioning (Chen et al., 1 May 2026). The approximate-category approach depends on algebraic group actions, truncated neighborhood algebras, and finite-field complexity results for fixed $F:\mathcal{C}\to\mathbf{Poset}$18 (Derksen, 2010).

Several limitations recur. Constructing the symmetric simplicial complex $F:\mathcal{C}\to\mathbf{Poset}$19 is combinatorially expensive for large hyperedge cardinality, so the hypergraph experiments truncate lifting at threshold $F:\mathcal{C}\to\mathbf{Poset}$20 (Choi et al., 6 Feb 2026). In CCWL, top-dimensional uniqueness can obstruct boundary recovery for isolated top cells, and the framework remains a color-refinement scheme, so families with extreme symmetries may require higher-order or global invariants beyond CCWL’s locality (Chen et al., 1 May 2026). In the approximate-category setting, the existence of a universal bounded depth $F:\mathcal{C}\to\mathbf{Poset}$21 that solves graph isomorphism in polynomial time remains open, as do questions about robustness across characteristics and coefficient growth over $F:\mathcal{C}\to\mathbf{Poset}$22 (Derksen, 2010).

A final misconception is that categorical enrichment automatically means using more neighborhoods or more structure. The papers show the opposite in a precise sense. CatWL proves equivalence of restricted adjacency sets under mild boundary-size conditions, and CCWL proves that upper and lower adjacencies already attain the expressive power of the full framework. The categorical contribution is therefore not maximal structural proliferation, but principled control over which structures are necessary, invariant, and sufficient.