1191dct-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. References1192shulman-2016shulman-2016.xmlCategorical logic from a categorical point of viewReference2016Michael Shulmanhttps://mikeshulman.github.io/catlog/catlog.pdf1194cruttwell-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.htmlBacklinks1197mat-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 theoryRelated1198dct-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.