1084dct-000Gdct-000G.xmlMonoidal promonadsDoctrine555#270unstable-270.xmlMotivationIn the standard way of defining a Markov category, one first postulates a monoidal category (with extra structure) whose morphisms are regarded as stochastic maps, then one carves out the deterministic maps as the morphisms satisfying a certain equation (commuting with copy maps). However, this logic can be turned around: start by postulating a monoidal category of deterministic maps, then introduce an identity-on-objects functor into another category of stochastic maps along with a compatible monoidal structure. The latter approach is axiomatized by a monoidal promonad.Like a promonad is equivalently an identity-on-objects functor, a monoidal promonad is equivalently an identity-on-objects strict monoidal functor (cf. Román 2024, Remark 8.2). However, that description suggests that one must explicitly give the associators and unitors of the target monoidal category. A virtue of the following definition is that the associators and unitors are only defined once, for the base monoidal category, and are then reused via the left and right actions of the promonad.556#271unstable-271.xmlMonoidal promonadsDefinitionA monoidal promonad consists of an (unbiased) monoidal category a promonad on the underlying category an (unbiased) monoidal product on , namely a natural transformation that obeys the following axioms: Compatibility of products: for any list of morphisms in , Functorality with respect to composition of arbitrary morphisms in Associativity and unitality relative to the associators and unitors of the base monoidal category .In the following theory for monoidal promonads, we generalize from the list double monad to an arbitrary lax double monad as the semantics. We can then recover not only monoidal promonads, but the more important symmetric monoidal promonads and other variants as models.557thy-000Gthy-000G.xmlMonoidal promonadsTheoryThe theory of (generalized) monoidal promonads is the modal double theory that combines the theory of generalized monoidal categories and the theory of promonads with a cell subject to the axioms: Compatibility of products: Functorality: , i.e., Associativity: Unitality: We can generalize still further by allowing the base category and the promonad to have generalized monoidal products with respect to different monads. See: Relative monoidal promonads.References1085roman-2024roman-2024.xmlString diagrams for physical duoidal categoriesReference2024Mario Román2406.19816Backlinks1087mat-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 theory1088dct-000Hdct-000H.xmlRelative monoidal promonadsDoctrine697#269unstable-269.xmlMotivationWe motivated the concept of a monoidal promonad by reconstructing the concept of a Markov category, starting from the category of deterministic maps. But a monoidal promonad does not account for an important distinction: the category of deterministic maps is not just a monoidal category but a cartesian monoidal category, whereas the promonad of stochastic maps, while it inherits copying and deleting maps, is not cartesian but only a semicartesian monoidal category . This motivates a further generalization of (generalized) monoidal promonads, where the two generalized monoidal products are with respect to different monads, related by a monad morphism. We call this structure a relative monoidal promonad (nonstandard terminology).698thy-000Hthy-000H.xmlRelative monoidal promonadsTheoryThe theory of relative monoidal promonads is the modal double theory , where is a monad morphism, that combines the theory of generalized monoidal categories (for the monad ) and the theory of promonads with a cell subject to the axioms: Compatibility of products: Functorality: Associativity: where the equation is well-formed by the compatibility of with the monad multiplications Unitality: where the equation uses the compatibility of with the monad unitsRelated1089dct-000Idct-000I.xmlMarkov categoriesDoctrine586#266unstable-266.xmlContrast with the standard definition of Markov categoryWarningA Markov category is standardly defined to be a symmetric monoidal category with a supply of commutative comonoids (also known as "copy-discard category"), such that the monoidal unit is a terminal object (Fritz 2020). While fairly succinct, this definition has some strange properties: It postulates an unnatural transformation , the family of copying maps, always questionable in category theory and simply not allowed in our approach to formal category theory, in which every transformation is automatically natural. In the original motivating example of a Markov category, the category of measurable spaces and Markov kernels, the equational characterization of determinism is subtly wrong (Patterson 2020, Corollary 3.2.2). The following definition corrects both problems by making the deterministic maps be structure rather than property. Be warned that it is not equivalent to, but slightly generalizes, the standard definition of a Markov category.587#267unstable-267.xmlMarkov categoriesDefinitionA Markov category is a model of the theory of relative monoidal promonads in the semantics , where the monad morphism is the inclusion, whose object components are identities.Thus, a Markov category consists of a cartesian monoidal category, whose morphisms are called deterministic, over which there is a semicartesian monoidal promonad, whose morphisms are called stochastic.588#268unstable-268.xmlEquivalent definitionsRemarkOther modal theories with the equivalent models can be given. For example, inspired by the unbiased adjoint characterization of cartesian categories, we could use the theory of right adjoints to the unit for the base category. The associator needed in the theory of relative monoidal promonads would then be derived rather than primitive: 1090dct-0004dct-0004.xmlPromonadsDoctrineA promonad is a monad in the bicategory of profunctors. Its interest lies in being equivalent to a pair of categories with an identity-on-objects functor from one to the other. Thus, a promonad is a categorical structure with two kinds of morphisms, where one can be cast to the other. Promonads can be iterated to capture hierarchies of morphisms with more than two levels.503thy-0004thy-0004.xmlPromonadsTheoryA theory of promonads is the simple double theory generated by an object a proarrow a globular cell , the unit subject to the equations Composition: Unitality: .504#250unstable-250.xmlRemarks on the axiomsRemarkIn graphical terms, the two unitality axioms are: The axioms imply the functorality equation via two possible calculations, for example: 505#289unstable-289.xmlModels of the theory of promonadsPropositionA model of this theory is precisely a promonad. 506#290unstable-290.xmlProofSee the blog post Patterson 2024. 1091dct-000Hdct-000H.xmlRelative monoidal promonadsDoctrine697#269unstable-269.xmlMotivationWe motivated the concept of a monoidal promonad by reconstructing the concept of a Markov category, starting from the category of deterministic maps. But a monoidal promonad does not account for an important distinction: the category of deterministic maps is not just a monoidal category but a cartesian monoidal category, whereas the promonad of stochastic maps, while it inherits copying and deleting maps, is not cartesian but only a semicartesian monoidal category . This motivates a further generalization of (generalized) monoidal promonads, where the two generalized monoidal products are with respect to different monads, related by a monad morphism. We call this structure a relative monoidal promonad (nonstandard terminology).698thy-000Hthy-000H.xmlRelative monoidal promonadsTheoryThe theory of relative monoidal promonads is the modal double theory , where is a monad morphism, that combines the theory of generalized monoidal categories (for the monad ) and the theory of promonads with a cell subject to the axioms: Compatibility of products: Functorality: Associativity: where the equation is well-formed by the compatibility of with the monad multiplications Unitality: where the equation uses the compatibility of with the monad units