October/Novemberfest Titles and Abstracts

Speaker: Thomas W Thorbjørnsen

Title: Finitely Adequate Modules in Synthetic Algebraic Geometry

Abstract:
Synthetic algebraic geometry (SAG) is an extension of homotopy type theory that provides a language for internal reasoning in the big Zariski topos. In SAG, we postulate the existence of a generic local ring R with some additional properties. Schemes over R are not defined by giving an underlying space a structure sheaf; rather, being a scheme is a property of the space itself. Sheaves of modules on a scheme can then be expressed simply as bundles of R-modules over it.

To find a notion for quasi-coherent sheaves internally, one might first try taking sheaves valued in finitely presented R-modules. However, the generic local ring R is provably not Noetherian, and so the category of finitely presented R-modules is not closed under kernels. Consequently, the cohomology groups of such sheaves may no longer be finitely presented. A natural next step is to take the kernel closure of the finitely presented R-modules, whose objects we call finitely adequate R-modules. Even though the cohomology of sheaves of finitely adequate modules may still fail to be finitely adequate, this class of modules exhibits many remarkable algebraic properties. By considering the geometry of these modules, we will show that they form a full, extension-closed abelian subcategory of the category of all R-modules.

Speaker: Paul Taylor

Title: Replacement as an Adjoint Functor Theorem

Abstract: Category Theory is a mature subject.  We have rewritten most of 20th century mathematics in the language of an elementary topos and we have studied other foundational systems formulated using categorical properties.  We do not need Set Theory, we should not formulate our subject in set-theoretic language and we should certainly not allow set theorists to tell us what our foundations are.

However, there are constructions, in particular infinite iteration of functors, that go outside topos logic but can supposedly be founded in ZF.

What, in our native language, is the principle underlying such constructions?

Our native language is written in adjunctions, so how can we derive a new way of justifying the existence of adjoints using IZF?

It is sometimes said that Replacement provides “big” colimits, but before invoking a colimit we must formulate its diagram.  This may not be definable in topos logic.

However, it may be definable as the image under a putative left adjoint of a diagram in another category that is topos-definable.

Underlying this is an example of an unbounded predicate that is allowed in Zermelo set theory but not topos logic, namely that a particular morphism has a universal property.

Powerful adjunctions are related to recursive constructions but recursion need not terminate. To handle this, we introduce a notion of “well founded object”.

Speaker: Matthew Di Meglio

Title: Factoring in (the other) pre-Hilbert *-categories: A generalisation of Douglas’ lemma

Abstract: Douglas’ lemma is a foundational result in functional analysis relating order, range inclusion and factorisation of operators on Hilbert spaces. The standard proof uses the Cauchy completeness of Banach spaces several times in a seemingly essential way. My talk will introduce the notion of pre-Hilbert *-categories and prove a generalisation of Douglas lemma for pre-Hilbert *-categories. This generalisation is surprising because the definition of pre-Hilbert *-categories is purely algebraic and even finitary: no real numbers, norms or limits of Cauchy sequences are assumed. This instantly gives us Douglas’ lemma for unitary representations of groupoids, for self-dual Hilbert modules over W*-algebras, and for finite-dimensional inner product modules over ordered division *-rings.

Speaker: Matthias Fresacher

Title: Categorical representation of DRC-semigroups

Abstract: DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular $∗$-, and $∗$-regular semigroups. In this joint work with James East, Azeef Muhammed and Tim Stokes, we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular $∗$-semigroups of East and Muhammed. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.

Speaker: Yorgo Chamoun

Title : A compositional definition of automata

Abstract : Finite state automata are widely used and studied because they conveniently allow one to describe and manipulate regular languages over a fixed alphabet. They, however, suffer from a small inconvenience: this notion is not compositional. Indeed, there is a natural notion of
composition of automata, corresponding to the identification of the final states of one automaton with the initial states of another, and it is well-known that the language recognized by the composite automaton is not in general the concatenation of the two languages, unless we suitably “unfold” the automata, or use silent transitions in the composition. We, however, adopt a completely different, categorical point of view: in order to make automata theory compositional, we simply identify the categorical operations that are necessary to build all automata, and we consider the free category with these operations. More precisely, we consider the free loop-free traced symmetric monoidal category containing a qualitative bialgebra and the alphabet as endomorphisms. Then we characterize this category as a symmetric monoidal category whose objects are natural numbers, and whose morphisms from m to n are automata with m inputs and n outputs. In other words, we give the universal property of the category of automata. The category Aut of automata thus obtained is compositional with respect to the language, in the sense that we can define the operation that associates to an automaton the language it recognizes as a functor on Aut, which sheds new lights on this operation. In addition, we relate this axiomatization to classical automata with silent transitions, by showing that, if we suitably quotient those, we recover our automata. We also provide a similar universal characterization for the category of classical automata based on the notion of Frobenius algebra.

Speaker: Omar Dennaoui

Title: Nerve Theorem for Cyclic Multicategories

Abstract: The classical nerve theorem establishes that categories can be characterized as simplicial sets satisfying the Segal condition, providing a bridge between categorical and simplicial perspectives.

Given a simplicial set $X$, it is the nerve of a category if and only if the following Segal map

\[X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1\]

is an isomorphism for all $n\geq 2$.

This characterization has proven fundamental in higher category theory and has been extended to various algebraic structures. We establish analogous nerve theorems for cyclic multicategories, allowing one to profitably view these structures as presheaves on a category of planar trees. Here “cyclic” refers to a rotational symmetry on operations that blurs the distinction between inputs and outputs. Cyclic multicategories and operads play important roles in multivariable adjunctions, cyclic homology, graph complexes, and topological field theories, among other applications.

We employ the general frameworks for abstract nerve theorems developed by Leinster and Weber. Specifically, we apply the machinery of monads with arities of Berger, Melliès, and Weber, it provides a systematic method for computing minimal dense generators of categories of algebras over strongly cartesian monads. We instantiate this for monads on the category of cyclic multigraphs to establish the theorems; in the case of the free cyclic operad monad, this was anticipated in Patrick Elliott’s thesis.

Speaker: Aleksandar Ivanov

Title: Bistellar Data in Arbitrary Dimensions

Abstract: The orbifold construction is a procedure that constructs new TQFTs from old ones given a particular algebraic input. I will review aspects of the construction, and then I will present a framework that algorithmically generates the most general such algebraic data called bistellar data. The benefit of the framework is that it gives an immediate description of the algebraic data in any dimension, as well as a description of bistellar bimodules and all further iterated bimodules, which should serve as higher cells in a Morita-style higher category.

Speaker: Eleftherios Chatzitheodoridis

Title: Trees associated with unitary partition complexes

Abstract: In a recent paper, Heuts and Moerdijk develop a poset of trees whose classifying space is homotopy equivalent to the n-th partition complex, that is, the classifying space of the poset of partitions of a set with n elements. The latter poset features in work of Arone, Dwyer, and Lesh, and it finds a unitary analog introduced by Arone and Lesh. In joint work with Julie Bergner, Pedro Brunialti Lima de Andrade, and Josh Turner, we develop a topological poset of trees whose classifying space is homotopy equivalent to the n-th unitary partition complex, that is, the classifying space of the topological poset of partitions of n-dimensional complex space in pairwise orthogonal subspaces. Our work builds on the study of unitary partition complexes by Bergner, Joachimi, Lesh, Stojanoska, and Wickelgren, and it also entails some new results in the theory of topological categories that may be of independent interest.

Speaker: Jana K. Nickel

Title: Distributive laws for relative monad

Abstract: This talk is based on work in progress joint with JS Lemay.

Relative monads form a generalization of monads where the underlying functor may have distinct domain and codomain categories. This makes it possible to study monad constructions beyond the classical setting of endofunctors and thereby provides a convenient language for algebraic structures. Given two ordinary monads, a distributive law between them is a natural transformation satisfying specific compatibility conditions with the monad structures which express abstractly that the two monads distribute one over the other. The data of a distributive law is equivalent to the data of an extension of one monad to the Kleisli category of the other one. Motivated by this latter characterization, I will present a notion of a distributive law between relative monads and discuss to which extent this recovers the non-relative case. Moreover, I will illustrate this concept of a relative distributive law by an example involving the Kleisli and Eilenberg-Moore categories of ordinary monads.

Speaker: Khyathi Komalan

Title: Organising Compositionality in Classical Mechanics

Abstract: Suppose we fix two springs. There are many ways to assemble them: we might leave them separate, make them share a mass, or connect them in some other way. The components stay the same, while the resulting mechanical system changes. Where should this choice of composition live mathematically?

Baez, Weisbart and Yassine model open classical systems using augmented spans. In joint work with David Jaz Myers, I’ll describe a reformulation within the Libkind-Myers double categorical framework for systems and interactions, working with smooth spaces equipped with closed 2-forms. The aim is to keep the systems themselves separate from the choice of how they are assembled. Tensoring places systems side by side without yet connecting them. An interaction then specifies how the combined system is to be reorganised, with pullback carrying out the assembly.

I’ll illustrate this through spring systems, including two springs sharing a mass, and show how their Lagrangian data fits naturally into this picture. If time permits, I’ll end by discussing how the same framework might incorporate Hamiltonian evolution through time.

Speaker: Ariel Rosenfield

Title: Universal algebra perspectives on hyperstructures

Authors: Ariel E Rosenfield, Florian Warg, Tony Zorman

Abstract: Several categories of hyperstructures, including the hypergroups of Marty and the mosaics of Nakamura and Reyes, are realizable as categories of objects with extra structure within the category of complete atomic Boolean algebras. Moreover, hypergroups are exactly the total dagger Frobenius algebras in the category of relations. We conjecture that this latter correspondence is a special case of one between hypergroupoids and total dagger Frobenius monoids in an arbitrary regular category.

Speaker: Arghan Dutta

Title: Symmetric monoidal $(\infty,n)$-categories

Abstract: Symmetric monoidal $(\infty,n)$-categories have been considered in a variety of forms and from several perspectives, including Lurie’s commutative algebra objects and, for $(\infty,1)$-categories, coCartesian fibrations over finite pointed sets satisfying a Segal condition; Gepner and Haugseng’s iterated enrichment; and the special $\Gamma$-objects of Schommer-Pries, Calaque–Scheimbauer, and Johnson-Freyd.

In this talk, I will introduce a definition in terms of left double $(\infty,n)$-fibrations, building on the framework for double higher categories developed by Moser, Rasekh, and Rovelli. Working internally in a suitable model category of $(\infty,n)$-categories, we express the symmetric monoidal Segal condition through spans associated with the inert maps of finite pointed sets. I will then focus on the quasi-categorical case, where we prove a Quillen equivalence with Lurie’s presentation via coCartesian fibrations; in the strict one-dimensional case, our approach recovers the ordinary symmetric monoidal categories. Our comparison uses strings of coCartesian morphisms and a staircase construction to pass between the two descriptions. I will explain the ideas behind these constructions and how they relate the two Segal conditions for symmetric monoidality.

Speaker: Chad Nester

Title: Crossed Tiloids and Graded Monoidal Categories

Abstract: This talk concerns strict double categories with exactly one object, which I will call “tiloids”. More specifically, I am interested in tiloids that are “crossed”, by which I mean that they possess well-behaved “crossing cells”, which are in many ways analogous to the braiding of a symmetric monoidal category, but do not seem to have been studied systematically.

I will give a definition of “crossed tiloid”, provide a number of examples, and will explain their connection to a recently introduced notion of graded monoidal category. This is all based on joint work with Matthew Earnshaw, which is currently in preparation.

Speaker: Jean-Baptiste Vienney

Title: Uniqueness and non-uniqueness of deriving transformations

Abstract: A differential category is an additive symmetric monoidal category with an algebra modality, axiomatizing smooth functions, and a deriving transformation, axiomatizing the differentiation of these smooth functions. An algebra modality may or may not have both the nullary and the binary Seely isomorphisms. Lemay proved that if an algebra modality has both Seely isomorphisms, then it has at most one deriving transformation. It was then an open problem whether every algebra modality has at most one deriving transformation. We answer this question in the negative by constructing algebra modalities with infinite sets of deriving transformations. We also clarify the relations between the nullary Seely isomorphism, the binary Seely isomorphism, and uniqueness of deriving transformations (that is, the property for an algebra modality to have at most one deriving transformation). First, we refine Lemay’s theorem by removing the assumption of the nullary Seely isomorphism. Then, we show by means of various examples that the six combinations of absence or presence of nullary Seely isomorphism, binary Seely isomorphism, and uniqueness of deriving transformations, which are not already ruled out by the refined uniqueness theorem, can be realized. Based on preprint arXiv:2510.03953.

Speaker: Aaron Fairbanks

Title: The pullback product of monads

Abstract: If T and S are two monads on the same category, the composite functor TS need not be a monad. Nevertheless, the pullback of the natural transformations from T and S into TS, induced by the units of T and S, does have the structure of a monad (whenever this pullback exists). Moreover, this construction generalizes beyond monads to monoids in any monoidal category, and beyond monoids to models of any multicategory with a unique constant at every sort. Joint work with Sridhar Ramesh.

Speaker: Jacob Neumann

Title: Introduction to Paranatural Category Theory

Abstract: We can understand the property of naturality as a kind of uniformity condition: the components of a natural transformation behave coherently with respect to their parameter category (the domain of the functors in question). The purpose of “diagonal naturality” is to articulate a similar notion of parametrically uniform transformation between mixed-variance functors. Such a notion has important applications in the theory of programming languages. However, the standard notion of a “dinatural transformation” well-known to category theorists is not up to the task: not only do dinatural transformations not compose (therefore preventing the formulation or study of a mixed-variance analogue to functor/presheaf categories), but their naturality condition simply fails to articulate the right notion of uniformity.

In this talk, I detail several improvements on the notion of “dinaturality”—strong dinaturality, paranaturality, and isofibrant paranaturality—which address these issues. All these notions of “uniform transformation of mixed-variance functors” compose. Moreover, the “paranaturality” notions allow for exponentiation (making them adequate for aforementioned programming language theory applications), and isofibrant paranaturality also has a Yoneda Lemma, demonstrating that these are the appropriate mixed-variant analogues of natural transformations.

Speaker: Xanna Little Berg

Title: Higher-Order Concurrent Processes for Implementing Recursion Patterns

Abstract: In sequential functional programming, inductive data types encompass data structures such as lists, trees, and the natural numbers. Functions from these types often follow basic patterns of recursion. Higher-order functions, such as “folds” and “maps,” can be used to abstract these patterns. Dual to inductive data types are coinductive codata types. Codata can represent possibly infinite data, such as infinite lists and conatural numbers (which include infinity and count down towards zero). A basic pattern for functions to coinductive types is an “unfold” which is dual to a fold.

The Categorical Message Passing Language (CaMPL) is a functional-style concurrent programming language with a categorical semantics. Its core programming feature is message passing along typed communication channels between processes. Concurrent types in CaMPL are thus channel types rather than data types: they define the structure of the interaction on the channel. The concurrent analogue of inductive and coinductive data types are protocols and coprotocols. These are user-defined channel types that can specify possibly infinite interactions between the processes connected by the channel. In this talk, we will explore channel structures that are the concurrent analogues of lists and trees, and we will explore the concurrent analogue of a “fold” which is a higher-order process called a “drive.” This will include discussion of the categorical semantics and demonstrations of CaMPL programs.

Speaker: Theofanis Chatzidiamantis-Christoforidis

Title: Fixed Point Properties in Synthetic Homotopy Theory

Abstract: Motivated by homotopy type theory (HoTT), we say that a space X has the homotopy-invariant fixed point property if for every continuous self-map f of X, we can choose a point x of X and a path from f(x) to x, and the choices of point and path are both continuous in f. Working in HoTT, we will present classes of counterexamples and examples of spaces with this property, which involve a model-independent description of the homotopy groups of the space of maps into a pointed connected 1-type. We work constructively, therefore our results can be interpreted in any ∞-topos. We will also connect our work to a stronger fixed point property on topological spaces previously studied by Szymik.

This talk is based on joint work with Dan Christensen.

Speaker: Daniel Almeida

Title: A categorical model structure for generalized algebraic theories

Abstract: Generalized algebraic theories, introduced by Cartmell in the late 1970s, extend the framework of Lawvere theories by allowing the introduction of dependent sorts. This feature makes such theories convenient, for example, for axiomatizing (1-categories of) higher categorical structures in a type-theoretic language. If T is a generalized algebraic theory, two different kinds of models of T have been traditionally considered: a strict one, where contexts are interpreted as iterated families of sets, and a categorical one, which is a model of the underlying finite-limit sketch (or clan) of the syntactic category C(T). These two versions often match, but not always. This relates to a more general issue regarding generalized algebraic theories: they can be arranged into a strict 2-category, but the morphisms that preserve all available structure are too strict to directly encode the intended categorical semantics. This can be studied by constructing a Quillen model structure on the category of generalized algebraic theories whose homotopy bicategory correctly expresses the weaker semantics – that gives, in particular, a strictification result for models of cofibrant theories. Talk based  on https://arxiv.org/abs/2608.27667

Speaker: Diego Berrio

Title: A 2-monadic approach to pseudo symmetric multicategories.

Abstract: Multicategories provide a setting for working with multi-input or multilinear maps by generalizing both operads and symmetric monoidal categories. They were introduced in Homotopy Theory to study the multiplicative properties of algebraic $K$-theory constructions. A multicategory is symmetric when it has an action of the symmetric group on multilinear maps that permutes inputs. By studying the 2-monad for symmetric multicategories on a suitable 2-category of non-symmetric multicategories enriched in $\mathbf{Cat}$, we arrive at the definition of pseudo symmetric multicategories enriched in $\mathbf{Cat}$,  as well as that of pseudo symmetric multifunctors between them. Pseudo symmetric multicategories have an action of the symmetric group on multilinear maps but only up to coherent isomorphisms, and pseudo symmetric multifunctors preserve this action up to coherent isomorphisms. We establish partial coherence results for pseudo symmetric multicategories and multifunctors, as well as new examples coming from algebraic $K$-theory. These examples lead to applications to multiplicative algebraic $K$-theory and lay the groundwork for future applications of our work to the equivariant homotopy theory setting. This is work in progress.

Speaker: J.S. Lemay

Title: The problem with Cartesian Fermat Categories and how to fix them (hopefully!)

Abstract:  We introduced Cartesian Fermat Categories (CFC) as a non-Lawvere theory analogue of a Fermat theory and showed they were Cartesian Differential Categories (CDC). Unfortunately, it turns out that our definition was too strong since not every Fermat theory is a CFC. In this talk, I will propose a way to fix the definition of a CFC. This is joint work with Carlos Pascual Miralles and also recaptures some of Ben Macadam’s previous work.