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Weak cartesian properties of simplicial sets

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Abstract

Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category \(\Delta \) to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in \(\Delta \) to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in \(\Delta \), which we characterize completely, along with several other classes of squares in \(\Delta \). Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.

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Notes

  1. It is worth noting that the theory of \((\infty ,1)\)-categories, which is modeled by quasicategories, also has closely related algebraic models [17].

  2. See for example [19, Section 14.2].

  3. Although the diagram still commutes when \(i = j - 1\), it is then no longer a pushout, as the single nontrivial \(\vee \)-component of its span is one of the following:

    figure w

  4. See e.g. [1], or [16, Chapter 13] for a textbook account.

  5. It is known that Graham reduction can be performed in any order, i.e. it is impossible to get stuck.

  6. Note that the assumption of connectedness guarantees that the set-theoretic intersection \(T_k \cap \bigcup _{i=1}^{k-1}\) is nonempty.

  7. A category is gaunt if the only isomorphisms are the identities.

  8. See property 5 of Theorem 13.2 in there.

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Acknowledgements

We first of all thank Joachim Kock for detailed comments on an earlier version, which have resulted in various improvements to the exposition. This paper originates from the Applied Category Theory 2019 school. We thank the organizers Daniel Cicala and Jules Hedges for having made it happen, the Computer Science Department of the University of Oxford for hosting the event, as well as all other participants of the school for the interesting discussions and insights, especially Martin Lundfall.

Funding

Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. Research for the third author was partly funded by the Fields Institute (Canada), and by the AFOSR Grants FA9550-19-1-0113 and FA9550-17-1-0058 (USA). The fourth author was supported by the National Defense Science and Engineering Graduate Fellowship Program.

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Authors and Affiliations

  1. Mansfield College, Oxford, UK

    Carmen Constantin

  2. Department of Mathematics, University of Innsbruck, Innsbruck, Austria

    Tobias Fritz

  3. Department of Computer Science, University of Oxford, Oxford, UK

    Paolo Perrone

  4. University of Virginia, Charlottesville, VA, USA

    Brandon T. Shapiro

Authors
  1. Carmen Constantin
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  2. Tobias Fritz
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  3. Paolo Perrone
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  4. Brandon T. Shapiro
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Correspondence to Brandon T. Shapiro.

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Communicated by George Janelidze.

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Constantin, C., Fritz, T., Perrone, P. et al. Weak cartesian properties of simplicial sets. J. Homotopy Relat. Struct. 18, 477–520 (2023). https://doi.org/10.1007/s40062-023-00334-1

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