442mdt-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 .References444shulman-2016shulman-2016.xmlCategorical logic from a categorical point of viewReference2016Michael Shulmanhttps://mikeshulman.github.io/catlog/catlog.pdfContext446mdt-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.