Fall 2025

Category theory offers powerful constructions such as the Yoneda lemma, Kan extensions, and Day convolution which naturally extend to enriched category theory: the objects, morphisms, and 2-morphisms behave sufficiently like categories, functors, and natural transformations to support these tools. In this perspective, both the 2-category of categories and that of V-enriched categories should arise as instances of a more general setting. Accordingly, formal category theory seeks to isolate the intrinsic properties of the 2-category Cat that make these constructions possible, and to determine the minimal structure a 2-category must carry to support them. 

While there already exist frameworks for formal category theory, we suggest one that is particularly suited for objects endowed with rich structure such as iteratively internal operads and their algebras. Here the 2-category Λ abstracting the 2-category Cat is explicitly equipped with an internal analogue of the category of sets -- an object S of Λ over which the objects of Λ are enriched, so as to obtain an internal analogue C(_, _) : Cᵒᵖ × C → S of the hom-functor. 

I will give an overview of this framework guided by the motivating example of the presheaf operad of the operad of trees, along with the resulting operadic nerve-realisation adjunction.

Control systems are dynamical systems which have a variable left "open" for a controller function to be plugged in. Control theory is the art of designing control functions such that the composite system exhibits some desired behavior. Control Lyapunov functions and control barrier functions certify the stability and safety respectively of control systems: if you can construct one of these functions for your system, then there exists a controller which will cause the system to exhibit the desired property. In this talk, we generalize these results to "control-coalgebras". A control-coalgebra of an endofunctor F on a category C generalizes the classical notion of control system, which are exactly control coalgebras of the tangent bundle functor on the category of manifolds. 

Quantified relevant logic has long intrigued philosophers, inspiring the development of new semantic frameworks—from enriched relational and neighborhood semantics for quantifiers to two-sorted algebraic structures. In this talk, we present an algebraic interpretation of quantified relevant logic RQ using polyadic De Morgan monoids. We establish a functional representation theorem for locally finite polyadic De Morgan monoids of infinite dimension. Finally, we discuss why this approach provides a legitimate framework by relating it to Andrew Tedder’s MG-structure. This work is a joint collaboration with Nicholas Ferenz.

Codd's Theorem, a fundamental result of database theory, asserts that relational algebra and relational calculus have the same expressive power on relational databases. We explore Codd's Theorem for databases over semirings and establish two different versions of this result for such databases: the first version involves the five basic operations of relational algebra, while in the second version the division operation is added to the five basic operations of relational algebra. In both versions, the difference operation of relations is given semantics using semirings with monus, while on the side of relational calculus a limited form of negation is used. The reason for considering these two different versions of Codd's theorem is that, unlike the case of ordinary relational databases, the division operation need not be expressible in terms of the five basic operations of relational algebra for databases over an arbitrary positive semiring; in fact, we show that this inexpressibility result holds even for bag databases.

- G. Badia, P. G. Kolaitis, C. Noguera, Codd's Theorem for Databases over Semirings, arxiv.org/abs/2501.16543

There are many existing dualities for bounded lattices. Some dualities rely on representations via sets of filters, and some rely on representations via sets of filters and sets of ideals. In this talk, I'll discuss a duality for bounded lattices that is based on the filter-ideal space of a lattice, which is the product of its space of filters and its space of ideals. This duality relies on three facts. First, the category of bounded lattices can be embedded into a category of Galois connections on some free distributive lattices. Second, the Priestley duals of these distributive lattices are order-homeomorphic to filter-ideal spaces of bounded lattices, and they arise from two Vietoris constructions on Priestley spaces. Third, via Priestley duality, Galois connections on distributive lattices correspond to certain relations on Priestley spaces. All of these facts already appear, in some way or other, in the literature. Taken together, however, they yield a new representation of bounded lattices as Priestley spaces endowed with a closed relation, in which lattice homomorphisms correspond to order-continuous functions which preserve the closed relation in a natural way.

Almost lattices are a generalization of lattices where associativity and commutativity for join are replaced by the weaker identities ((x ∨ y) ∨ z) ∧ w = (x ∨ (y ∨ z)) ∧ w and (x ∨ y) ∧ z = (y ∨ x) ∧ z, and in addition associativity for meet and the following absorption laws are assumed: x ∧ (x ∨ y) = x, x ∨ (x ∧ y) = x and (x ∧ y) ∨ y = y. A preposet is a structure (A, ≤, ⪯) consisting of a nonempty set X with a preorder ⪯ and a partial order ≤ which refines ⪯ and satisfying the formula x, y ≤ z and x ⪯ y and y ⪯ x ⇒ x = y. We prove that almost lattices are equivalent to preposets in which a notion of weak infimum and weak supremum can be defined for all pairs of elements. As a consequence we conclude that the noncommutative meet operation in an almost distributive lattice is uniquely determined by the structure of the preposet. This is joint work with Peter Jipsen and based on similar results by Michael Kinyon for left-handed binormal skew lattices.

It is well known that distributive lattices are an abstraction of families of sets closed under the operations of union and intersection. On sets of functions, the operations of override and restriction can be viewed as noncommutative generalizations of union and intersection, which motivates the theory of almost distributive lattices (ADLs) due to Swamy and Rao (1981). In 2013 Bauer, Cvetko-Vah, Gehrke, van Gool and Kudryavtseva developed a noncommutative Priestley duality based on sheaves over sets for certain skew lattices that are equivalent to associative ADLs with 0. In joint work with Gezahagne Addis we show that this duality can be adapted to associative ADLs in general, as well as to associative ADLs with 1 and with both 0 and 1.

- A. Bauer, K. Cvetko-Vah, M. Gehrke, S. van Gool and G. Kudryavtseva, A non-commutative Priestley duality, Topology and its Applications, 160 (2013), 1423-1438, http://dx.doi.org/10.1016/j.topol.2013.05.012

- J.E. Leech, Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond, Univ. of Primorska Press, 2020, https://doi.org/10.26493/978-961-293-027-1 

- U.M. Swamy and G.C. Rao, Almost distributive lattices, J. Austral. Math. Soc. (Series A) 31, (1981), 77-91, https://doi.org/10.1017/S1446788700018498 

In this talk, we first provide plenty of examples (finite as well as infinite) of almost distributive lattices (ADLs), supported by Hasse diagrams, exhibiting different ADL properties. Moreover, we present some congruence properties of the class of ADLs. Continuing our investigation on finite ADLs, we obtain an inductive algorithm to describe the cardinality of some finite ADL L in terms of the cardinality of a distributive lattice that is embedded in L.