arXiv subjects

Simon Cho

Publications and source records attributed to Simon Cho.

3 recordsLinked to original sources

arXiv2019

Quantales, persistence, and magnitude homology

We construct a nerve functor parametrized by a choice of quantale, exhibiting both the Vietoris-Rips complex and the magnitude nerve as instances of this nerve for different choices of monoidal structure on $\mathbb{R}$. Furthermore, the difference between how persistent homology processes the Vietoris-Rips complex and how magnitude homology processes the magnitude nerve is cast as a choice of whether or not to "localize" the corresponding nerves along $\mathbb{R}$ in a precise sense. Lastly, we mention some application-oriented observations naturally suggested by the perspective mentioned above.

math.AT↗

arXiv2019

Categorical semantics of metric spaces and continuous logic

Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this "continuous semantics" is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval [0,1] in the category of metric spaces as a "continuous subobject classifier" giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic. Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.

math.CT↗

arXiv2016

An application of continuous logic to fixed point theory

In aiming to apply to a broader class of examples the Avigad-Iovino "ultraproducts and metastability" approach to obtaining uniformity for convergence of sequences, we construct a framework using continuous logic that in particular is able to handle discontinuous functions in its domain of discourse. This setup weakens the usual continuity requirements for functions, but compensates for the loss of control by introducing a notion of "linear structure" that captures in a quite general way the situation of having geodesics between every pair of points, and has as a special case the vector space structure of Banach spaces. We use this to apply the Avigad-Iovino method to specific convergence results from functional analysis involving iterations of discontinuous functions, and so obtain uniform metastable convergence in those results.

math.LO↗