924dct-000Fdct-000F.xmlGeneralized multicategoriesDoctrineDepending on the choice of monad, a generalized multicategory could be a multicategory, a symmetric multicategory, a cartesian multicategory, or something more exotic. Our axiomatization of these doctrines as a modal double theory is reminiscent of Cruttwell-Shulman's definition but bypasses the complicated construction of the "horizontal Kleisli virtual double category" of a monad on a virtual double category.The generalized multicategories axiomatized here are "normalized," in Cruttwell-Shulman's jargon, but see also the unnormalized notion.646thy-000Fthy-000F.xmlGeneralized multicategoriesTheoryThe theory of generalized multicategories is the modal virtual double theory generated by an object a proarrow having the composites: Composition: The composite exists and is the restriction . Normalization: The unit is the restriction . Associativity: The composite exists and is the restriction . and satisfying the cell equations: Unitality on the left: Unitality on the right: Associativity: 421#249unstable-249.xmlInterpretation of the axiomsRemarkThe axioms of this theory might seem opaque at first glance, but the three axioms on composites are intuitively just transposes of familiar laws. The first two are transposes of the monad algebra laws, which we might write informally as commutative diagrams: Similarly, the third axiom on composites can be informally pictured as the naturality square for the transposed monad multiplication at : The three equational axioms then ensure that any "extra" composition cells are uniquely determined by the multicategory's composition operation.The two axioms labeled "associativity" are somewhat mysterious. As the name suggests, they are used in proving that the multicategory has associative composition. Taken together, they can be read as saying that the cell component of the monad multiplication, when transformed by sliding arrows into proarrows via companions (cf. Patterson 2026, Equation 2.1), is a composition (opcartesian) cell. Note that so constraining the monad on the theory does not constrain the monad on the semantics; it only determines the image of a composition cell in models of the theory. Open question: Should such laws belong to the definition of a modal virtual double theory?647#272unstable-272.xmlModels of the theory of generalized multicategoriesPropositionA model of the modal theory of generalized multicategories in , where is a virtual equipment and is a monad in VDCs, is precisely a normalized -multicategory.In particular, a model... ...in is exactly a multicategory ...in is exactly a symmetric multicategory ...in is exactly a cartesian multicategory. 648#273unstable-273.xmlProofThe proof is the same as in the unnormalized case, except that is now a restriction cell, which makes the category in be isomorphic to the multicategory's underlying category of unary morphisms. References925double-transposing-2026double-transposing-2026.xmlTransposing cartesian and other structure in double categoriesReference2026Evan Patterson2404.0883510.1016/j.jpaa.2026.108233927cruttwell-shulman-2010cruttwell-shulman-2010.xmlA unified framework for generalized multicategoriesReference2010G.S.H. CruttwellMichael A. Shulman0907.2460http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.htmlBacklinks930mat-0001index.xmlCatColab: Mathematical Documentation529#251unstable-251.xmlIntroductionAs the name suggests, CatColab is based on mathematical ideas from category theory. It is a specific design goal that the system be usable without any knowledge of such ideas. These docs are for people who want to know about the mathematics, perhaps because they want to develop the core package catlog or just out of curiosity. These pages are currently far from being a self-contained introduction, serving mainly to collect new bits of CatColab-related mathematics as they are created. For proper exposition, see the published literature.In mathematical terms, CatColab is an editor for categorical structures and their morphisms and higher morphisms. The meta-logical framework organizing these structures is based on double category theory. More precisely, the domain-specific logics in CatColab are defined by double theories, and the models in CatColab are models of double theories.The library of domain-specific logics in CatColab is inspired by a wide body of research in applied category theory and beyond. Incomplete bibliographies are in these docs and the catlog docs.530#252unstable-252.xmlDoctrinesSystems of categorical logic, or doctrines, whose free objects have a systems interpretation: Doctrine Free objects Systems interpretation Categories Graphs [numerous] Profunctors Bridges between graphs Database schemas Signed categories Signed graphs Regulatory networks/causal loop diagrams (CLDs) Delayable signed categories Signed graphs with delays CLDs with delays Nullable signed categories Nullable signed graphs CLDs with indeterminates Categories with links Graphs with links Stock-flow diagrams Symmetric (or commutative) monoidal categories Hypergraphs Petri nets/reaction networks Multicategories Multigraphs Term graphs/diagrammatic equations Doctrines whose objects have interpretations as process theories: Doctrine As process theories Promonads / Graded categories Hierarchy of morphisms Generalized monoidal categories / Generalized PROs / Unnormalized PROs Monoidal theory, algebraic theory, ... Multicategories / Generalized multicategories / Unnormalized multicategories / Representable multicategories Operad, algebraic theory, ... Cartesian categories Algebraic theory Monoidal promonads / Relative monoidal promonads Hierarchy of processes Markov categories Probabilistic processes/statistical models Categories of partial maps Fallible processes/processes with constraints Partial Markov categories Fallible, probabilistic processes Additional doctrines: Companions, Conjoints, Fibrations, Categories with pullbacks531dbl-0001dbl-0001.xmlDouble category theoryDouble category theory: TabulatorsDouble-categorical logic: Double Lawvere theories Discrete models and free models Modal double theories Bibliography: Double category theory931dct-0005dct-0005.xmlGeneralized monoidal categoriesDoctrineMonoidal categories can be axiomatized as double theories in several ways. Lambert & Patterson 2024 present cartesian double theories for both strict and weak monoidal categories, categorifying the familiar Lawvere theory for monoids. Here we present modal double theories for "generalized" monoidal categories parameterized by a monad, which are unbiased in the same spirit as generalized multicategories and generalized PROs.440thy-0005thy-0005.xmlMonad algebras and pseudoalgebrasTheoryThe theory of monad algebras is the one-dimensional theory of a monad algebra, viewed as a double theory the tight direction. Thus, it is the modal double theory generated by an object , the carrier an arrow , the structure map subject to the equations Associativity: Unitality: The theory of monad lax algebras is the modal double theory generated by an object , the carrier an arrow , the structure map two special cells, the associator and unitor: These cells obey three coherence axioms, omitted here but stated in the notes Patterson 2025.Finally, the theory of monad pseudoalgebras is the theory of monad lax algebras augmented with two more special cells, and , along with equations making them loose inverses to and , respectively.When the intended semantics is the list double monad on or one of its many variants, we'll give these theories more suggestive names. We'll also denote the structure map as a tensor product, . Generic name Name for semantics in list monads theory of monad algebras theory of generalized strict monoidal categories theory of monad pseudoalgebras theory of generalized monoidal categories theory of monad lax algebras theory of generalized lax monoidal categories 441#288unstable-288.xmlModels of the theory of generalized monoidal categoriesPropositionA model of the modal theory of generalized (strict) monoidal categories... ... in is exactly a (strict) monoidal category ... in is exactly a symmetric (strict) monoidal category ... in is exactly a cartesian (strict) monoidal category ... in is exactly a cocartesian (strict) monoidal category where in all cases the monoidal product and additional structure is unbiased.932dct-000Edct-000E.xmlGeneralized PROsDoctrineA single-sorted PRO is a strict monoidal category whose monoid of objects is freely generated by a distinguished object. Similarly, a single-sorted PROP (resp. Lawvere theory ) is a symmetric (resp. cartesian) strict monoidal category whose objects are freely generated by a distinguished object.Here we axiomatize the multi-sorted versions of these doctrines via a modal double theory. The formulation is possibly novel, though it's in the spirit of Cruttwell-Shulman's framework for generalized multicategories. Just as with multicategories, generalized PROs can be normalized or unnormalized; we take "generalized PROs" to be normalized by default, even if the unnormalized notion is more basic in our axiomatization.642thy-000Ethy-000E.xmlGeneralized PROsTheoryThe theory of generalized PROs is the theory of unnormalized PROs augmented with the normalization axiom: the unit cell is a restriction, i.e., the unique globular cell factoring through any chosen restriction is an isomorphism.643#274unstable-274.xmlModels of the theory of generalized PROsPropositionA model of the modal theory of generalized PROs... ...in is exactly a PRO ...in is exactly a PROP ...in is exactly a Lawvere theory where in all cases we mean the multi-sorted version. 644#275unstable-275.xmlProofFollows from the corresponding statement about unnormalized PROs along with the normalization axiom, which makes the identity-on-objects functor into an isomorphism of categories. 933dct-000Odct-000O.xmlRepresentable multicategoriesDoctrineAbstracting the universal property of the tensor product of vector spaces, a tensor product of a list of objects in a multicategory is an object along with a universal multimorphism . A representable multicategory is a multicategory in which every list of objects admits a tensor product.Abstracting still further, representability can be defined for any generalized multicategory.686thy-000Othy-000O.xmlRepresentable multicategoriesTheoryThe theory of representable multicategories extends the theory of generalized multicategories with the generators an arrow a nullary cell and the follwing relations: Unitality: the nullary cell uniquely determined by the universal property of restriction as shown below is loosely invertible (that is, there is a generating nullary cell mutually inverse to ). Universal property: the composite cell is tightly invertible (that is, there is a generating globular cell mutually inverse to the globular cell above).630#247unstable-247.xmlComparison with literatureRemarkThis double theory abstracts, with only minor variations, the standard definition of representability for an ordinary multicategory (e.g., Shulman 2016, Definition 2.2.3). In their framework for generalized multicategories, Cruttwell-Shulman define representability differently (CS10, Section 9) so as to make it a straightforward theorem that representable generalized multicategories are equivalent to generalized monoidal categories (pseudo -algebras). I have not investigated the relationship between these two definitions.687#255unstable-255.xmlModels of the theory of representable multicategoriesPropositionA model of the modal theory of representable multicategories in is precisely a representable multicategory equipped with a choice of tensor products. Similar statements hold for and symmetric multicategories, and for and cartesian multicategories. 688#256unstable-256.xmlProof Consider the case of ordinary multicategories; the others are similar. A model consists of a multicategory with underlying profunctor , a functor , and a family of multimorphisms natural in lists of morphisms of , with the universal property that for every choice of lists of lists of objects, precomposition with gives a bijection Moreover, by the unitality axiom, for each object in , the morphism in corresponding to the unary multimorphism is an isomorphism. Thus, the standard definition of representability is recovered by applying the universal property when all but one of the lists are singletons. 934dct-000Ndct-000N.xmlUnnormalized multicategoriesDoctrineIn the Cruttwell-Shulman framework for generalized multicategories, the main definition, that of a -monoid for a monad on a virtual double category (CS10, Definition 4.3), often yields not only a multicategory but the further data of a category equipped with an identity-on-objects functor into the multicategory's underlying category of unary morphisms. While it has found a few applications (Baez-Dolan 1998, Cheng 2004), this extra data is usually unwanted. It can be removed by "normalization," leading to Cruttwell-Shulman's preferred definition of a "generalized multicategory" as a normalized -monoid (CS10, Definition 8.19).We also take generalized multicategories to be normalized by default. However, it is still instructive to axiomatize unnormalized multicategories, as it makes for a crisper analogy with promonads.651thy-000Nthy-000N.xmlUnnormalized multicategoriesTheoryThe theory of unnormalized multicategories is the modal virtual double theory generated by an object a proarrow a cell, the unit having the composites: Composition: The composite exists and is the restriction . Associativity: The composite exists and is the restriction . and satisfying the cell equations: Unitality on the left: Unitality on the right: Associativity: 652#248unstable-248.xmlComparison with theory of promonadsRemarkThis theory might also be called the "theory of -promonads" as it's analogous to the theory of promonads except that the monad acts on the source of the proarrow . The axioms on composition and unitality correspond to the three axioms of the theory of promonads. The remaining two axioms, on associativity, uniquely extend the right action from (hence from ) and enable another way to decompose a length-three composite of multimorphisms. In fact, if one forces to be the identity monad, then the two associativity axioms trivialize and the theory of promonads is recovered.653#257unstable-257.xmlModels of the theory of unnormalized multicategoriesPropositionA model of the modal theory of unnormalized multicategories in , where is a virtual equipment and is a monad in VDCs, is precisely an unnormalized -multicategory.In particular, a model in consists of a category, a multicategory, and an identity-on-objects functor from the category into the multicategory's underlying category of unary morphisms. Similar statements hold for and symmetric multicategories, and for and cartesian multicategories. 654#258unstable-258.xmlProof A model consists, first of all, of a category object in and a profunctor . By the composition axiom, we can define a cell in as: The image of this cell in defines the composition operation of the multicategory; conversely, by the universal property of restriction, choosing the multicategory's composition operation uniquely determines the image of the composition cell on the right-hand side. Next, the image of the unit cell furnishes a natural transformation in : The functorality equation in (abbreviated using the right unitality axiom) holds, as shown by precomposing both sides with the nullary composition cell and using its universal property. Thus, preserves composition (cf. Cheng 2004, Definition 2.1, Axiom 6) and hence is an identity-on-objects functor from to the unary morphisms of , where by definition the identity unaries in are the images under of the identities in . Moreover, the left and right actions of the profunctor are uniquely determined by this functor, due to the corresponding unitality axioms. The unitality laws of the generalized multicategory follow from the unitality of the profunctor actions. It remains to prove the associativity law of the generalized multicategory. First, note that the composition cell for and generates no extra data as its image is uniquely determined by the other data, using the final axiom and the universal property of restriction. By the associativity of composites, the assumption that preserves composites and restrictions, and the axioms on composites, we have the equation Postcompose this equation by the restriction Then, on the left-hand side, apply the equation which follows from preserving restrictions and the formula for restrictions in a virtual equipment (CS10, Theorem 7.16). Meanwhile, on the right-hand side, apply the equation which follows from the second associativity axiom and (CS10, Theorem 7.16) again. Deduce the associativity law for a generalized multicategory: Related935dct-000Adct-000A.xmlMulticategoriesDoctrineA multicategory , also known as a colored or typed non-symmetric operad, is precisely a model of the following double theory.936thy-000Athy-000A.xmlPromonoidsTheoryThe theory of promonoids is the cartesian double theory generated by an object equipped with proarrows , subject to associativity and unitality conditions. In other words, there is a unique counary proarrow for every . See (Lambert & Patterson 2024, Theory 6.9).Arbitrary proarrows must then be a product of proarrows for some partition of . Thus the loose category of is the augmented simplex category , the category of totally-ordered, possibly-empty finite sets and order-preserving maps between them. This phenomenon is familiar from the fact that the augmented simplicial category is itself the strict monoidal category freely generated by a monoid object. Compare Pisani 2025 for a theory of multicategories in which the proarrows are instead maps of sets with orders on the fibers only, for better comparison with symmetric multicategories.A model of the theory of promonoids maps the object to the object set of the multicategory and maps the proarrow to the span whose preimage at is the set of multimorphisms .937dct-000Ndct-000N.xmlUnnormalized multicategoriesDoctrineIn the Cruttwell-Shulman framework for generalized multicategories, the main definition, that of a -monoid for a monad on a virtual double category (CS10, Definition 4.3), often yields not only a multicategory but the further data of a category equipped with an identity-on-objects functor into the multicategory's underlying category of unary morphisms. While it has found a few applications (Baez-Dolan 1998, Cheng 2004), this extra data is usually unwanted. It can be removed by "normalization," leading to Cruttwell-Shulman's preferred definition of a "generalized multicategory" as a normalized -monoid (CS10, Definition 8.19).We also take generalized multicategories to be normalized by default. However, it is still instructive to axiomatize unnormalized multicategories, as it makes for a crisper analogy with promonads.651thy-000Nthy-000N.xmlUnnormalized multicategoriesTheoryThe theory of unnormalized multicategories is the modal virtual double theory generated by an object a proarrow a cell, the unit having the composites: Composition: The composite exists and is the restriction . Associativity: The composite exists and is the restriction . and satisfying the cell equations: Unitality on the left: Unitality on the right: Associativity: 652#248unstable-248.xmlComparison with theory of promonadsRemarkThis theory might also be called the "theory of -promonads" as it's analogous to the theory of promonads except that the monad acts on the source of the proarrow . The axioms on composition and unitality correspond to the three axioms of the theory of promonads. The remaining two axioms, on associativity, uniquely extend the right action from (hence from ) and enable another way to decompose a length-three composite of multimorphisms. In fact, if one forces to be the identity monad, then the two associativity axioms trivialize and the theory of promonads is recovered.653#257unstable-257.xmlModels of the theory of unnormalized multicategoriesPropositionA model of the modal theory of unnormalized multicategories in , where is a virtual equipment and is a monad in VDCs, is precisely an unnormalized -multicategory.In particular, a model in consists of a category, a multicategory, and an identity-on-objects functor from the category into the multicategory's underlying category of unary morphisms. Similar statements hold for and symmetric multicategories, and for and cartesian multicategories. 654#258unstable-258.xmlProof A model consists, first of all, of a category object in and a profunctor . By the composition axiom, we can define a cell in as: The image of this cell in defines the composition operation of the multicategory; conversely, by the universal property of restriction, choosing the multicategory's composition operation uniquely determines the image of the composition cell on the right-hand side. Next, the image of the unit cell furnishes a natural transformation in : The functorality equation in (abbreviated using the right unitality axiom) holds, as shown by precomposing both sides with the nullary composition cell and using its universal property. Thus, preserves composition (cf. Cheng 2004, Definition 2.1, Axiom 6) and hence is an identity-on-objects functor from to the unary morphisms of , where by definition the identity unaries in are the images under of the identities in . Moreover, the left and right actions of the profunctor are uniquely determined by this functor, due to the corresponding unitality axioms. The unitality laws of the generalized multicategory follow from the unitality of the profunctor actions. It remains to prove the associativity law of the generalized multicategory. First, note that the composition cell for and generates no extra data as its image is uniquely determined by the other data, using the final axiom and the universal property of restriction. By the associativity of composites, the assumption that preserves composites and restrictions, and the axioms on composites, we have the equation Postcompose this equation by the restriction Then, on the left-hand side, apply the equation which follows from preserving restrictions and the formula for restrictions in a virtual equipment (CS10, Theorem 7.16). Meanwhile, on the right-hand side, apply the equation which follows from the second associativity axiom and (CS10, Theorem 7.16) again. Deduce the associativity law for a generalized multicategory: