1153dct-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. References1154promonads-post-2024promonads-post-2024.xmlPromonads are algebrasReference2024129Evan Pattersonhttps://topos.institute/blog/2024-01-29-algebras-are-promonads/Backlinks1156mat-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 theory1157dct-000Kdct-000K.xmlGraded categoriesDoctrineStrictly speaking, graded categories are not a doctrine but a family of doctrines. For each monoidal category , there is a doctrine of -graded categories. See the blog post Patterson 2025 and references therein. Note that here we depart from that post and the literature by making the grading in covariant rather than contravariant.480thy-000Kthy-000K.xmlGraded categoriesTheoryGiven a strict monoidal category , the theory of -graded categories is the delooping double category . That is, it is the simple double theory with trivial category of objects and arrows, and proarrows and cells given by the objects and morphisms of , respectively. External composition in is the monoidal product in .481#263unstable-263.xmlPromonadsExampleA promonad is a category graded by the linear order , made into a monoidal poset under the join.1158dct-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.1159dct-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: