914dct-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. References915cruttwell-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.htmlBacklinks918mat-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 theory919dbl-0008dbl-0008.xmlDouble Lawvere theoriesA multisorted Lawvere theory is a finite-product theory with a distinguished set of generating objects/types. This idea can be formulated precisely in a couple of ways; here is Todd Trimble's definition.416dbl-0009dbl-0009.xmlMultisorted Lawvere theoryDefinitionGiven a set , a Lawvere theory with sorts is a category with finite products equipped with an identity-on-objects, finite-product preserving functor Just as a finite-product double theory is a categorification of a finite-product theory, so should a "double Lawvere theory" be a categorification of a Lawvere theory. Since we think of a double theory as having both object types (its objects) and morphism types (its proarrows), a multisorted double Lawvere theory ought to have both object sorts and morphism sorts. But stating this definition is more complicated than simply replacing the family construction with the double family construction, because the source or target of a morphism sort must be a finite family of object sorts rather than a single sort. For now we make the following incomplete definition.417dbl-000Adbl-000A.xmlMultisorted double Lawvere theory (WIP)DefinitionGiven a set , a double Lawvere theory with object sorts is a finite-product double theory equipped with an identity-on-objects, finite-product preserving functor 920dct-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.921dct-000Pdct-000P.xmlUnnormalized PROsDoctrineAnalogous to an unnormalized multicategory, an unnormalized PRO consists of a category, a (multisorted) PRO, and an identity-on-objects functor from the category to the PRO's underlying category of single-input, single-output morphisms. Unnormalized PROPs and Lawvere theories are defined similarly.680thy-000Pthy-000P.xmlUnnormalized PROsTheoryThe theory of unnormalized PROs is the modal double theory generated by an object a proarrow a pair of cells, the product and the unit: subject to the equations Composition: Functorality of product: , i.e., Associativity of product: Unitality of product: Unitality of unit: 681#246unstable-246.xmlComparison with theory of promonadsRemarkThe part of this theory comprising the object , proarrow , and cell is analogous to the theory of promonads except that the monad acts on the source and the target of .Here too the functorality axiom for the unit can be derived from the other axioms. To do so, precompose, say, the left equation of the unitality axiom with and use the naturality of at along with unitality of the product to deduce that .682#253unstable-253.xmlModels of the theory of unnormalized PROsPropositionA model of the theory of unnormalized PROs in consists of a category, a PRO, and an identity-on-objects functor from the category to the PRO's underlying category of unary and counary morphisms. Similar statements hold for and PROPs, and for and Lawvere theories. 683#254unstable-254.xmlProof We sketch the proof for bare PROs; the other cases are similar. A model of the theory consists, first of all, of a category and a profunctor equipped with a composition . The objects of and the heteromorphisms of are, respectively, the sorts and the morphisms of the PRO. In addition, there is an (unbiased) tensor product and a transformation . The (derived) functorality equation for implies that is an identity-on-objects functor, while the two unitality axioms for ensure that the left and right actions of the profunctor are fully determined by the composition on . The remaining axioms then simply assert that is a strict monoidal category. Related922dct-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. 923dct-000Pdct-000P.xmlUnnormalized PROsDoctrineAnalogous to an unnormalized multicategory, an unnormalized PRO consists of a category, a (multisorted) PRO, and an identity-on-objects functor from the category to the PRO's underlying category of single-input, single-output morphisms. Unnormalized PROPs and Lawvere theories are defined similarly.680thy-000Pthy-000P.xmlUnnormalized PROsTheoryThe theory of unnormalized PROs is the modal double theory generated by an object a proarrow a pair of cells, the product and the unit: subject to the equations Composition: Functorality of product: , i.e., Associativity of product: Unitality of product: Unitality of unit: 681#246unstable-246.xmlComparison with theory of promonadsRemarkThe part of this theory comprising the object , proarrow , and cell is analogous to the theory of promonads except that the monad acts on the source and the target of .Here too the functorality axiom for the unit can be derived from the other axioms. To do so, precompose, say, the left equation of the unitality axiom with and use the naturality of at along with unitality of the product to deduce that .682#253unstable-253.xmlModels of the theory of unnormalized PROsPropositionA model of the theory of unnormalized PROs in consists of a category, a PRO, and an identity-on-objects functor from the category to the PRO's underlying category of unary and counary morphisms. Similar statements hold for and PROPs, and for and Lawvere theories. 683#254unstable-254.xmlProof We sketch the proof for bare PROs; the other cases are similar. A model of the theory consists, first of all, of a category and a profunctor equipped with a composition . The objects of and the heteromorphisms of are, respectively, the sorts and the morphisms of the PRO. In addition, there is an (unbiased) tensor product and a transformation . The (derived) functorality equation for implies that is an identity-on-objects functor, while the two unitality axioms for ensure that the left and right actions of the profunctor are fully determined by the composition on . The remaining axioms then simply assert that is a strict monoidal category.