arXiv subjects

Thomas Jan Mikhail

Publications and source records attributed to Thomas Jan Mikhail.

4 recordsLinked to original sources

arXiv2025

A type-theoretic definition of lax $(\infty,\infty)$-limits

We introduce and study a purely syntactic notion of lax cones and $(\infty,\infty)$-limits on finite computads in \texttt{CaTT}, a type theory for $(\infty,\infty)$-categories due to Finster and Mimram. Conveniently, finite computads are precisely the contexts in \texttt{CaTT}. We define a cone over a context to be a context, which is obtained by induction over the list of variables of the underlying context. In the case where the underlying context is globular we give an explicit description of the cone and conjecture that an analogous description continues to hold also for general contexts. We use the cone to control the types of the term constructors for the universal cone. The implementation of the universal property follows a similar line of ideas. Starting with a cone as a context, a set of context extension rules produce a context with the shape of a transfor between cones, i.e.~a higher morphism between cones. As in the case of cones, we use this context as a template to control the types of the term constructor required for universal property.

math.CT↗

arXiv2025

Abacus bicomodule configurations and the Bergner-Osorno-Ozornova-Rovelli-Scheimbauer equivalence

A theorem of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer states an equivalence between 2-Segal spaces and certain augmented stable double Segal spaces. In this paper we establish more general equivalences, involving simplicial maps of 2-Segal spaces and abacus bicomodule configurations, extending results of Carlier. The BOORS equivalence is recovered from the special case of the identity map. One main ingredient is an analysis of the relationship between the BOORS and Carlier notions of augmentation, hitherto considered unrelated.

math.CT↗

arXiv2025

Free loop spaces and the Cauchy--Frobenius Lemma

We upgrade the Cauchy--Frobenius Lemma (`Burnside's Lemma') to a homotopy equivalence of $\infty$-groupoids, essentially given by double counting/Fubini in the free loop space of the quotient.

math.AT↗

arXiv2024

Fuzzy simplicial sets and their application to geometric data analysis

In this article, we expand upon the concepts introduced by David Spivak about the relationship between the category $\mathbf{UM}$ of uber metric spaces and the category $\mathbf{sFuz}$ of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions. We generalize the adjunction between $\mathbf{UM}$ and $\mathbf{sFuz}$, derive an explicit description of colimits in $\mathbf{UM}$, and show that $\mathbf{UM}$ can be embedded into $\mathbf{sFuz}$. Furthermore, we prove analogous results for the category of extended-pseudo metric spaces $\mathbf{EPMet}$. We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and $\mathbf{sFuz}$, which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods (source code on github). Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets, demonstrate how our method improves upon them and provide explanations for observed differences.

math.AT↗