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Joseph Helfer

Publications and source records attributed to Joseph Helfer.

9 recordsLinked to original sources
arXiv2026

Set-theoretic universes and paradoxes in 2-topoi

This paper continues the development of (elementary) 2-topos theory, a foundational theory based on an axiomatization of the 2-category of categories. We prove that any 2-topos contains a model of intuitionistic ZF set theory, and we use this to show that, if appropriate size restraints are not imposed, then a Burali-Forti type paradox can be deduced from the 2-topos axioms. The set-theoretic universe is produced as a special case of a general construction giving an internal category of models in a 2-topos of an arbitrary finite higher-order theory, which in turn is carried out using a notion of "topos sketch". As a byproduct of our construction, we also make contact with the subject of "algebraic set theory", introducing a novel approach to the construction of set-theoretic universes from a "category of classes".

math.CT↗
arXiv2025

Y is a least fixed point combinator

The theory of recursive functions is related in a well-known way to the notion of *least fixed points*, by endowing a set of partial functions with an ordering in terms of their domain of definition. When terms in the pure lambda-calculus are considered as partial functions on the set of reduced lambda-terms, they inherit such a partial order. We prove that Curry's well-known fixed point combinator Y produces least fixed points with respect to this partial order.

math.LO↗
arXiv2025

The integral chow ring of $M_2^{ct}$

This paper computes the integral Chow ring of the moduli space $M_2^{ct}$ of stable genus 2 curves of compact type. This is done by excising boundary strata from $\bar M_2$ one-by-one. During this process, we determine the Chow rings of all other open strata in $\bar M_2$ with $Z[1/2]$-coefficients.

math.AG↗
arXiv2024

Internal 1-topoi in 2-topoi

We further develop the notion of elementary 2-topos, introduced by Weber, by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an appropriate sense. The axioms introduced for this purpose are closure conditions on the DOFs having "S-small fibres". Among these closure conditions, the most interesting one asserts that a certain DOF, analogous to the "subset fibration" over Set, has small fibres. The remaining new axioms concern "groupoidal" objects in a 2-category, which are seen to play a significant role in the general theory. We prove two results to the effect that a 2-category C satisfying these axioms is "determined by" its groupoidal objects: the first shows that C is equivalent to a 2-category of internal categories built out of groupoidal objects, and the second shows that the groupoidal objects are dense in C.

math.CT↗
arXiv2023

Severi curves of rational elliptic surfaces

We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their geometric genus using quasi-modular forms. We conjecture that these Severi curves are eventually reducible, and we formulate a precise conjecture for their degrees in $\mathbb{P}^2$, featuring a divisor sum formula for collision multiplicities of branch points.

math.AG↗
arXiv2021

Exotic tight contact structures on $\mathbb{R}^n$

We introduce a variant of contact homology for convex open contact manifolds. As an application, we prove the existence of (in fact, infinitely many) exotic tight contact structures on $\mathbb{R}^{2n-1}$ for all $n>2$.

math.SG↗
arXiv2019

First-order homotopical logic

We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from arXiv:1905.10690 that any Grothendieck fibration of the kind being considered can automatically be upgraded to a 2-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into 2-dimensional fibrations.

math.LO↗
arXiv2019

Homotopies in Grothendieck fibrations

We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the slice categories $\mathbf{C}/A$, and we show that in the case $\mathbf{C}=\mathbf{Top}$, our construction applied to this fibration recovers the usual 2-category of spaces.

math.CT↗