arXiv subjects

Graham Manuell

Publications and source records attributed to Graham Manuell.

23 records · Page 2Linked to original sources

arXiv2021

2-dimensional bifunctor theorems and distributive laws

In this paper we consider the conditions that need to be satisfied by two families of pseudofunctors with a common codomain for them to be collated into a bifunctor. We observe similarities between these conditions and distributive laws of monads before providing a unified framework from which both of these results may be inferred. We do this by proving a version of the bifunctor theorem for lax functors. We then show that these generalised distributive laws may be arranged into a 2-category Dist(B,C,D), which is equivalent to Lax(B,Lax(C,D)). The collation of a distributive law into its associated bifunctor extends to a 2-functor into Lax($B \times C$, D), which corresponds to uncurrying via the aforementioned equivalence. We also describe subcategories on which collation itself restricts to an equivalence. Finally, we exhibit a number of natural categorical constructions as special cases of our result.

math.CT↗

arXiv2021

A simple lower bound for ARRIVAL

The ARRIVIAL problem introduced by Dohrau, Gärtner, Kohler, Matoušek and Welzl concerns a train moving on a directed graph proceeding along outward edges according to the position of 'switches' at each vertex, which in turn are toggled whenever the train passes through them. The problem asks whether the train every reaches a designated destination vertex. It is known that ARRIVAL is contained in UP $\cap$ coUP, while the previously best published lower bound is that it is NL-hard. In this note we provide a simple reduction to the $\mathsf{DIGICOMP}_\mathsf{EXP}$ problem considered by Aaronson. It follows in particular that ARRIVAL is both CC-hard and PL-hard.

cs.CC↗

arXiv2019

The congruence biframe as a quasi-uniform bicompletion

Künzi and Ferrario have shown that a $T_0$ space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictly zero-dimensional biframe is a congruence biframe if and only if it is bicomplete in the same quasi-uniformity. As a corollary we obtain a new proof of a result of Plewe that a congruence frame is ultraparacompact. The main result makes use of a new construction of the bicompletion of a quasi-uniform biframe as a quotient of the Samuel compactification.

math.GN↗

arXiv2017

Strictly zero-dimensional biframes and a characterisation of congruence frames

Strictly zero-dimensional biframes were introduced by Banaschewski and Brümmer as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal monomorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.

math.GN↗

arXiv2017

A special class of congruences on $κ$-frames

Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a $κ$-frame need not be Boolean. We characterise these so-called d-reduced $κ$-frames as those which may be embedded as a generating sub-$κ$-frame of a Boolean frame. We introduce the notion of the closure of a $κ$-frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove $κ$-frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced $κ$-frames are precisely the quotients of $κ$-frames by clear congruences and that every $κ$-frame congruence is the meet of clear congruences.

math.RA↗