Victor Bloch
Stochastic Processes as Simplicial Markov Kernels
Simplicial sets provide a combinatorial model of topological spaces. They consist of abstract simplices of varying dimensions, subject to compatibility conditions, which describe how the simplices fit together. Interestingly, these compatibility conditions admit a probabilistic interpretation; they can express the consistency requirements that determine how a stochastic process is specified over time. Motivated by Markov categories -- a categorical framework for probability theory, inspired by the compositionality of Markov kernels -- we introduce "simplicial Markov kernels", which allow stochastic processes to be described as certain simplicial maps. This raises a natural question: to what extent can this category-theoretic approach recover the classical, measure-theoretic theory of stochastic processes? This is the central question of my ongoing research.
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Clémence Chanavat
Diagrammatic (∞, n)-categories.
The diagrammatic model of (∞, n)-categories is an emerging model of higher categories defined as certain sheaves on regular and directed CW-complexes. This model natively supports a wide variety of operations, including joins, Gray tensor products, suspensions, as well as inductive and coinductive notions of equivalence. It is also the first model of higher categories to satisfy a semi-strictification theorem: every diagrammatic (∞, n)-category is equivalent to one whose composition laws are algebraic and satisfy strict exchange and associativity.
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Benachir El Allaoui
Ext-vanishing for functor categories over semi-additive categories
This poster studies functor categories whose source is semi-additive. The main result is a vanishing theorem for higher Ext groups between linearizations of additive functors. Since the category of additive monoid-valued functors is not abelian, the proof replaces the usual Dold–Kan approach by simplicial projective resolutions and reduces the computation to the homology of simplicial commutative monoids.
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Rubén Izquierdo López
Higher form symmetries and general conservation laws
In physics, the Noether correspondence between higher-form symmetries (symmetries parametrized by closed forms) and more general conservation laws (lower-degree forms that are closed on solutions) has attained importance in both classical and quantum theory. This correspondence is well known among physicists, but lacks a fully formalized mathematical framework. In this contribution, I will present how higher-form symmetries can be understood in the classical setting as a special class of multivector fields on a multisymplectic manifold, and how the associated conservation laws arise as Hamiltonian forms corresponding to these multivectors.
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Luis Kattwinkel
From Lawvere Theories to Operadic Algebras
Lawvere theories can be seen either as categories of affine geometric spaces or as certain compatible collections of operations - a rather tautological instance of the usual geometry-algebra correspondence. One of the most compelling properties of geometric theories built from Lawvere theories is that their 1st order infinitesimal data may be elegantly accessed via the higher categorical machine called stabilization. We use recent advances in Goodwillie calculus, the higher order expansion of this machine, to investigate how category theory extracts infinitesimal data of arbitrary order from Lawvere theories and stores it in the format of operadic algebras.
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Yun Liu
Quasi-flag manifolds and moment graphs
The equivariant rational cohomology of flag manifolds is closely related to the algebra of quasi-invariants and quasi-covariants, which are generalizations of invariant polynomials for finite reflection groups. In this poster, we will explain how to obtain a mod-p cohomological decomposition (except for p=2) of a flag manifold along its moment graph, and how togeneralize it to construct quasi-flag manifolds. These results are obtained by constructing rational models of the new spaces in terms of coaffine stacks — derived stacks introduced by Toën and Lurie as an algebro-geometric framework for rational homotopy theory — which are derived version of the ordinary varieties of quasi-invariants, obtained from the same construction in the $\infty$-category of coaffine stacks rather than in affine schemes.
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Santiago Pareja Pérez
An axiomatic cheatsheet for graphical calculus
This poster serves as both an introduction to the general principles behind graphical calculus and a cheatsheet of the axioms for some categories with extra structure (such as symmetric, braided, pivotal, and ribbon categories). As an application, we sketch the construction of Reshetikhin–Turaev invariants of 3-manifolds.
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Vasileios Patzalis
Bounded Cohomology and Bounded Products
Bounded cohomology of groups and spaces is a refinement of ordinary cohomology with deep connections to differential geometry, topology, and group theory. We study bounded cohomology of simplicial sets, viewed as an object of the abelian category W, the heart of the canonical t-structure on D(Ban). This refinement has better formal properties, particularly with respect to colimits. Motivated by the comparison with ordinary cohomology, we construct a refined bounded product functor in W and prove that bounded products commute with cohomology, unlike in the classical seminormed setting. As applications, we obtain a Milnor short exact sequence and a formulation of the Serre spectral sequence in bounded cohomology without additional technical assumptions.
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Jan Pulmann
Higher Quantum Odd Symplectic Category
The symplectic “category” is a category with partially defined composition, well suited for symplectic geometry. In odd symplectic geometry, a natural quantization was introduced by Ševera, involving half-densities and generalizing the Batalin-Vilkovisky fiber integral. I will describe a linear version of this category and discuss possibilities for introducing higher cells. Based on joint works with B. Jurčo and M. Zika.
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Pablo Sánchez Martínez
A weak stratified homotopy equivalence from a simplicial complex and a perversity
We obtain a weak stratified homotopy equivalence between the perverse stratification of a simplicial complex and the classifying space of the corresponding perverse poset in the framework first constructed by Stephen Nand-Lal and then further studied by Lukas Waas in their respective theses. We can then distinguish this framework from Douteau's, since this is not a weak stratified homotopy equivalence in his model structure for stratified spaces. We hope to further study the relations between the constructible derived categories associated to these spaces from the obtained weak equivalence.
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Maximilian Stegemeyer
Transport functions for principal bundles and Morse homology with DG coefficients
We construct a Morse-theoretical way of describing principal fiber bundles over a closed manifold via a functor from the Morse flow category to the category induced by a topological group. These 'transport functions' can be used to set up a Morse complex with differential graded coefficients in the style of Barraud-Damian-Humilière-Oancea. The homology of this complex computes the singular homology of an associated bundle. Transport functions behave particularly well for Lie groups and we study some functoriality properties of this construction.
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Andy Sukowski-Bang
Colimits of Enriched Categories
We introduce categories of strings to give an explicit hom-object formula for colimits of enriched categories. As corollaries, we recover the rigidification of simplicial sets via Dugger and Spivak’s necklace construction and describe colimits in $\mathrm{Cat}$ and of Lawvere metric spaces.
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Hao Xu
Classical Symmetry TFTs for Continuous Symmetries via Higher Symplectic Geometry
We propose a shifted-symplectic formulation of a classical continuous analogue of the symmetry TFT paradigm. Let $G$ be an algebraic or Lie group acting by topological defects on an $n$-dimensional classical topological sigma model with target an $(n-1)$-shifted symplectic derived stack $(X,\omega)$ via the AKSZ construction. We argue that the corresponding $(n+1)$-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack $T^∗[n](\mathrm B G)$, equivalently the $(n+1)$-dimensional BF theory for $G$. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in $T^∗[n] (\mathrm B G)$. We realize the gauging of the $G$-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, syplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous SymTFTs. In this refinement, higher gerbes on $\mathrm B G$ encode classical analogues of 't Hooft anomaly data by decorating the shifted cotangent bulk and its Lagrangian boundary conditions. Finally, in dimension three we compare the infinitesimal BF model $\mathrm B(\mathfrak g\ltimes\mathfrak g^\vee)$ with the factorizable double $\mathrm B(\mathfrak g\oplus \mathfrak g)$. The resulting topological boundaries are described by Lagrangian Lie subalgebras, and the factorizable case relates the SymTFT dictionary to $r$-matrices and Belavin--Drinfeld data.