nLab Open problems in topos theory

This page is to record the reference

• William Lawvere:

  
  

  Open problems in topos theory

  
  

  “For Martin Hyland and Peter Johnstone in honor of their sixtieth birthdays”

  (April 2009)

  [pdf, pdf]

stating seven problems in topos theory, unsolved at time of writing:

First problem

Concerning the existence of a proper class of quotient toposes of a Grothendieck topos (with ‘quotient maps’ defined to be ‘connected surjective geometric morphisms’).

Solution for the case of Boolean toposes: Hora 2023

A special example in the non-Boolean case: Hora & Kamio 2024a

Full solution: Kamio & Hora 2024b

Third problem

The syntactic effects of the boundary operator $\partial$ on geometric theories $\mathbb{T}$ as well as its semantic effects on its models are described in section 10.2 of Caramello 2018.

Fourth problem

To calculate the Aufhebung of the opposition between skeleta and coskeleta in the category of symmetric simplicial sets (the category of presheaves on the category of non-empty finite sets).

The solution by Hora & Maehara 2025 as well as the remarks concerning cyclic sets by Kennett, Riehl, Roy & Zaks 2011 suggests the following revised problem 4’:

Is the Aufhebungs relation of skew-simplicial sets more generally identical to the relation for SSet?

Or even to consider this question for crossed groups relative to Eilenberg-Zilber categories or generalized Reedy categories.

Seventh problem

Problem 7, “The algebra of time”, concerns the characterization of Toposes of laws of motion

References

• C. Kennett, Emily Riehl, Michael Roy, M. Zaks: Levels in the toposes of simplicial sets and cubical sets, JPAA 215 5 (2011) 949-961 [arXiv:1003.5944, doi:j.jpaa.2010.07.002]

• Olivia Caramello: Theories, Sites, Toposes, Oxford University Press (2018) [ISBN:9780198758914, preview:pdf]

• Ryuya Hora: Internal Parameterization of Hyperconnected Quotients [arXiv:2302.06851]

  talk at Category Theory conference (July 2023) [pdf, video]

• Ryuya Hora, Yuhi Kamio: Quotient toposes of discrete dynamical systems, Journal of Pure and Applied Algebra 228 8 (2024) 107657 [doi;10.1016/j.jpaa.2024.107657, arXiv:2310.02647]

• Yuhi Kamio, Ryuya Hora: A solution to the first Lawvere’s problem A Grothendieck topos that has a proper class many quotient topoi [arXiv:2407.17105]

• Ryuya Hora, Yuki Maehara: Lawvere’s fourth open problem: Levels in the topos of symmetric simplicial sets [arXiv:2503.03439]