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Pedro Solórzano

Publications and source records attributed to Pedro Solórzano.

11 recordsLinked to original sources
arXiv2026

Tininess and right adjoints to exponentials

Objects $T$ whose exponential functor $(-)^T$ admits a right adjoint $(-)_T$ are known under different names. The fact that they exist, yet that the only set that satisfies this in the category of sets is the singleton made Lawvere suggest they ought to be ``amazingly tiny'' -- hence Lawvere's acronym ``A.T.O.M.'' This report explores how intuitively tiny any such object is. Evidences both in favor and to the contrary are produced by looking at their categorical behavior (subobjects, quotients, retracts, etc) when the ambient category is a topos. The topological behavior (connectedness, contractibility, connected components, etc) of both $T$ and $(-)_T$ is further analyzed in toposes that satisfy certain precohesive conditions over their decidable objects, where this tininess is tested against parts of Lawvere's foundational proposal for Synthetic Differential Geometry.

math.CT↗
arXiv2025

Connectedness through decidable quotients

By looking at decidable quotients, a sufficient condition is provided to guarantee that (1) the full subcategory of decidable objects of a topos is an exponential ideal and that (2) the classical notion of connectedness for an object $X$ coincides with $ΠX=1$, where $Π$ is the left-adjoint functor of the inclusion of the decidable objects. The addition of this condition to McLarty's axiomatic set up for Synthetic Differential Geometry makes any topos that satisfies it precohesive over the topos of its decidable objects. A converse is also provided.

math.CT↗
arXiv2024

Naive homotopy theories in cartesian closed categories

An elementary notion of homotopy can be introduced between arrows in a cartesian closed category $E$. The input is a finite-product-preserving endofunctor $Π_0$ with a natural transformation $p$ from the identity which is surjective on global elements. As expected, the output is a new category $E_p$ with objects the same objects as $E$. Further assumptions on $E$ provide a finer description of $E_p$ that relates it to the classical homotopy theory where $Π_0$ could be interpreted as the ``path-connected components'' functor on convenient categories of topological spaces. In particular, if $E$ is a 2-value topos the supports of which split and is furthermore assumed to be precohesive over a boolean base, then the passage from $E$ to $E_p$ is naturally described in terms of explicit homotopies -- as is the internal notion of contractible space.

math.CT↗
arXiv2024

On logical parameterizations and functional representability in local set theories

There is a well-known inclusion $ι_\mathscr{E}$ of a topos $\mathscr{E}$ in the linguistic topos $\mathscr{T}(Σ)$ of its internal language $Σ$ that proves both toposes to be equivalent. There is also a canonical translation $η_S$ for any local set theory $S$ into the local set theory $Σ$ of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from $\mathscr{T}(S)$ to $\mathscr{T}(Σ)$. Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that $ι_{\mathscr{T}(S)}$ parameterizes $\mathscr{T}(η_S)$ in such a way that syntactic $S$-functions are represented by themselves in $Σ$.

math.LO↗
arXiv2020

Twistor sections of Dirac bundles

A Dirac bundle is a euclidean bundle over a riemannian manifold $M$ which is a compatible left $C\ell(M)$-module, together with a metric connection also compatible with the Clifford action in a natural way. We prove some vanishing theorems and introduce the twistor equation within this framework. In particular, we exhibit a characterization of solutions for this equation in terms of the Dirac operator $D$ and a suitable Weitzenböck-type curvature operator $\mathcal{R}$. Finally, we analyze the especial case of the Clifford bundle to prove existence of nontrivial solutions of the twistor equation on spheres.

math.DG↗
arXiv2015

Waning in principal bundles

Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a subbundle in a natural way. Berger 3-spheres provide an example where the limit fibers are still of dimension 3. The fibers are shown to be entirely determined by the riemannian holonomy of the chosen bi-invariant metric.

math.DG↗
arXiv2011

Convergence of vector bundles with metrics of Sasaki-type

If a sequence of Riemannian manifolds, $X_i$, converges in the pointed Gromov-Hausdorff sense to a limit space, $X_\infty$, and if $E_i$ are vector bundles over $X_i$ endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the $E_i$ converges in the pointed Gromov-Hausdorff sense to a metric space, $E_\infty$. The projection maps $π_i$ converge to a limit submetry $π_\infty$ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to $\R^k/G$, where $G$ is a closed subgroup of $O(k)$ ---called the {\em wane group}--- that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms $μ_i=\|\cdot\|_i$ converges to a map $μ_{\infty}$ compatible with the re-scaling in $\R^k/G$ and the $\R$-action on $E_i$ converges to an $\R-$action on $E_{\infty}$ compatible with the limiting norm. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g. when a single compact $n$-dimensional manifold is re-scaled to a point, the limit fiber is $\R^n/H$ where $H$ is the closure of the holonomy group of the compact manifold considered. An appropriate notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the $\R$-action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Uniqueness is not true in general, but a necessary condition is given in terms of the aforementioned wane groups $G$.

math.DG↗
arXiv2010

Holonomic Spaces

A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$, of $Aut(V, \|\cdot\|)$ and a group-norm, $L$, with a convexity property. We prove that with the metric $d_L(u,v)=\inf_{a\in H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$, $V$ is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle $E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than the subspace topology while still preserving many desirable properties. Using these notions, we introduce the notion of holonomy radius for a Riemannian manifold and prove it is positive. These results are applicable to the Gromov-Hausdorff convergence of Riemannian manifolds.

math.DG↗
arXiv2009

Submanifolds and the Sasaki Metric

This is the content of a talk given by the author at the 2009 Lehigh University Geometry/Topology Conference. Using the definition of connection given by Dieudonné, the Sasaki metric on the tangent bundle to a Riemannian manifold is expressed in a natural way. Also, the following property is established. The induced metric on the tangent bundle of an isometrically embedded submanifold is the Sasaki metric if and only if the submanifold is totally geodesic.

math.DG↗