Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1501.05691

A Note on the Uniform Kan Condition in Nominal Cubical Sets

Robert Harper·Kuen-Bang Hou
View original source ↗Read source document ↗Export metadata ↓

Abstract

Bezem, Coquand, and Huber have recently given a constructively valid model of higher type theory in a category of nominal cubical sets satisfying a novel condition, called the uniform Kan condition (UKC), which generalizes the standard cubical Kan condition (as considered by, for example, Williamson in his survey of combinatorial homotopy theory) to admit phantom "additional" dimensions in open boxes. This note, which represents the authors' attempts to fill in the details of the UKC, is intended for newcomers to the field who may appreciate a more explicit formulation and development of the main ideas. The crux of the exposition is an analogue of the Yoneda Lemma for co-sieves that relates geometric open boxes bijectively to their algebraic counterparts, much as its progenitor for representables relates geometric cubes to their algebraic counterparts in a cubical set. This characterization is used to give a formulation of uniform Kan fibrations in which uniformity emerges as naturality in the additional dimensions.

Explore related subjects

math.LOcs.PLmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robert Harper, Kuen-Bang Hou. 2015-01-23. A Note on the Uniform Kan Condition in Nominal Cubical Sets. https://arxiv.org/abs/1501.05691

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗