902dct-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.References903double-monads-notes-2025double-monads-notes-2025.xmlDouble monads (handwritten notes)Reference20256Evan Pattersonhttps://drive.google.com/file/d/1njl_alPhN35XW3Ev_A7Su8Cc6GfF2ofn905cartesian-double-theories-2024cartesian-double-theories-2024.xmlCartesian double theories: A double-categorical framework for categorical doctrinesReference2024Michael LambertEvan Patterson10.1016/j.aim.2024.1096302310.05384Backlinks908mat-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 theory909thy-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.Related910mdt-0008mdt-0008.xmlModal double theories2025728Kevin Carlson (after Evan Patterson)424mdt-0001mdt-0001.xmlDouble monadDefinition2025714Kevin CarlsonA (strict, pseudo, lax, or colax) double monad is a monad in the 2-category of double categories, strict, pseudo, lax, or colax double functors, and tight natural transformations.426mdt-0002mdt-0002.xmlThe free monoid double monadDefinition2025714Kevin CarlsonThe free monoid monad on extends, since (as a polynomial functor) it preserves pullbacks, to a pseudo double monad, also called , on the double category of sets and spans. Concretely, to apply to a span , you just compose this span, seen as a functor , with , and similarly for the unit and multiplication. (This is always how you get a colax double functor on spans out of a functor between categories with pullbacks; the pseudo-ness is equivalent to preservation of those pullbacks.)The example below records a substantial generalization of this construction.428mdt-0007mdt-0007.xmlThe generalized list monadsDefinition2025716Kevin CarlsonLet be a sub-double category of , the skeletal double category of finite sets. We must add a certain technical requirement on , called being a "faithful cartesian club" in , which amounts to requiring that the sum of a span-indexed family of spans in is still in (see the monad multiplication definition below), and is obviously satisfied in the examples about to be listed. To wit, our motivating examples are the following: The one-object full sub-double category spanned by , so that the spans are all those of the form . The double category of identity spans of finite sets. The double category of companions of bijections between finite sets, i.e. spans of the form The double category of companions of functions between finite sets, i.e. spans of the form The double category of conjoints of functions between finite sets, i.e. spans of the form The double category of all spans of finite sets (with one or both components injective, surjective, bijective), i.e. spans of the form In fact, for any of the five classes we can consider spans with either leg in any of the classes, making for examples in all.Consider a span of sets. We define as a certain span . Here, and will be the set , that is the set of all diagrams of the form whose first line is in . The projections onto are visible.We now use this construction to define an endofunctor of , called or for short. On the tight category, as defined above. On each span, as defined above.Given a map of spans as below and an element , we define to be the composite: This visibly produces a functor on the loose category of .For unitors, we embed the identity span into as the elements for , which visibly satisfies naturality of unitors.For laxators, given composable spans and composable elements as below, we define the laxator as the composite below: Associativity and the naturality of laxators now just follow from the well-definedness of pastings in . That leaves the unitality axiom, which says that pasting with the identity cell on a tight arrow is the identity operation. Thus is indeed a lax double endofunctor of .We now put a double monad structure on . The tight part of the monad will be restricted from that of , which implies that contains and the sum of any family of its elements indexed by any of its members. (The only obvious examples of that are all the natural numbers, the positive ones, and , although more singular examples exist.) On spans, the unit will map an element of a span to the element of . All the axioms of a strict natural transformation are quite tautological.For the multiplication at a span , the action on an element of uses the multiplication of pointwise: The "faithful cartesian club" condition we require on is precisely that the flattened span lies in , which as mentioned above is immediate when each side of is determined by lying in the identity, bijection, injective, surjective, or arbitrary functions. The naturality of multiplication in cells of is clear; we turn to the other two axioms of naturality of .Coherence of with external units says that the composite , where is the unitor, coincides with . The latter is easier to compute: it sends a list of lists to the constant span on its flattening, like so: The unitor sends a list of lists to the span below, which we see is indeed sent to the right place by : Finally, for coherence of laxators we must show that . Calculating again the right-hand side first, we see it has the following action on an element of : On the other hand, acts on such an as follows: The key observation to verify this last axiom is that distributivity of pullbacks in means that . We have thus verified that is a strict natural transformation.Since the unit and multiplication are both just tuples of the unit and multiplication of , the monad laws come for free, and we have established that is a lax double monad on .430mdt-0004mdt-0004.xmlWeaknessDefinition2025714Kevin CarlsonA level of weakness is an element of the poset of levels of weakness, whose elements signify "strict", "pseudo", "lax", and "colax" respectively. The order is given by .432mdt-0005mdt-0005.xmlAlgebra over a double monadDefinition2025714Kevin CarlsonIf are comparable levels of weakness and is a -double monad, then a -algebra over is a -double functor equipped with a tight natural transformation satisfying the usual algebra laws, i.e. and . (That is, it is a -module carried by in the 2-category .) Note that these are all "strict algebras" in the sense of the formal theory of monads, i.e. the algebra laws are satisfied on the nose. By default, we'll assume .If , then an algebra is just a (strict) algebra over the 2-monad induced by on the tight 2-category of . But by increasing we can get quite interesting algebras even for double monads whose base has a boring tight 2-category, like :If , then the carrier is a category object in , consisting of an endo-proarrow plus the unitor and the laxator satisfying associativity and unitality.The algebra structure has components and which constitute -algebra structures on and respectively.As with any natural transformation between lax functors out of , and together constitute a functor between the category objects. Thus we can see an -algebra over precisely as a -algebra structure on a category object in , carried by a functor.In particular, when the double monad is naturally induced by a pullback-preserving monad on a category with pullbacks, then is essentially just again, and an -algebra over is a category in with -algebra structures on objects and arrows such that source and target are algebra homomorphisms and the algebra maps comprise a functor . Thus while an -algebra over the free monoid monad on is a monoid, an -algebra is a strict monoidal category.434mdt-0003mdt-0003.xmlAlgebras over generalized list monadsExample2025714Kevin CarlsonWe consider -algebras over various generalized list monads . Note that any such algebra structure endows a category with a monoid structure on objects. On morphisms, maps any map in to a "tensor product" , such that given the monad multiplication situation we have the unit axiom, which always says the same thing it says for algebras of , and more interestingly the multiplication axiom . Let us turn to examples.When , we have for any two objects a distinguished morphism arising from applied to the diagram below, where is the nontrivial automorphism. Since is a functor and the span is an involution, we see that , which together with the monoid structure on given by restricting along the map begins to suggest that is a (strict) symmetric monoidal category structure on . Indeed we extract such a structure using external functoriality: by considering first applying to the diagram below and then composing, versus first composing and then applying , we find that , where is an atomic "twisted tensor" not present in the usual biased formulation of SMC's. By symmetry, we also have , which provides the naturality condition of the braiding. In fact there are exactly analogous natural "multi-braidings" corresponding to any permutation of , and even more, we have twisted tensors such that , where indicates composition in along a -twisted span. All the above follows from functoriality of , without yet touching its interaction with the monad multiplication , which is why we so far have only described the properties of braidings at a single fixed at a time. Of course, braiding by a permutation like ought to be the same as braiding by and tensoring with an identity, which is precisely what that last axiom gives us. For this particular case, consider the following situation, where is the span such that : If we compose the above cell with , we get the element of , which sends to . If instead we apply , we get the element of , which sends to . This produces the relationship between braidings at different levels we had been hoping for, which says more generally that, given any permutation of which fixes a certain partition, we can either braid morphisms locally on each piece of the partition, then tensor the result together, or we can first tensor the morphisms together and then braid the result, and the result is the same. Together with the functoriality results of the previous paragraph, this shows that all the operations of on morphisms are uniquely determined by the unbraided tensor product and by the binary braiding , so at least at the level of objects we have a notion of (strict) unbiased symmetric monoidal category equivalent to the usual one. For the remaining examples, we shall not go into so much detail on the unbiasing results. When , we get symmetric monoidal categories again, since oppositization maps the previous choice of onto this one. I assume is also just symmetric monoidal categories, but it's not immediately obvious how. gives cocartesian monoidal categories. The codiagonal and unit of an object arise from the spans: Much as in the previous case, we can check that these structures constitute a natural commutative monoid structure (commutativity using external functoriality), which gives the result by Fox's theorem. We can also construct cocartesian generalized "braidings", as needed in an unbiased notion of cocartesian monoidal category. Taking opposites, gives cartesian monoidal categories. Presumably adding bijections to either identity leg above doesn't change the result. gives biproduct categories. Indeed, by multiplication over the span we get a natural morphism for every object in ; thus, is equipped with zero morphisms, and more generally the commutative monoid and cocommutative comonoid structures combine to enrich in commutative monoids. As in any category enriched in commutative monoids and with products (or coproducts), the products are biproducts. We won't explore injections and surjections in depth here, but you get things like semicartesian monoidal categories, or "relevance" monoidal (where there are diagonals but no deletions), or their duals, from picking one or the other leg identity. The classes like , and so on should let you surgically remove your choice of the unit, counit, multiplication, or comultiplication from a biproduct category, though I have no examples of these guys nor do I know whether they are interesting. We could also soften not to be a wide subcategory, restricting then the monad on objects in one way or another. The only example I have in mind at the moment is to let be all the spans with both ends terminal, which gives a category enriched in (not necessarily commutative!) monoids. These aren't particularly well known, but they exist; for instance any homomorphism of monoids gives a monoid-enriched category on three objects with , and , and defining the nontrivial composition. Note though that there are some semi-trivial compositions here, since for instance rather than as one is used to from the monoidal closed case; all such compositions have codomain , though.912dct-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. 913dct-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.