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Keywords:
algebraic injective; globular theory; faithfulness conjecture
algebraic injective; globular theory; faithfulness conjecture
Summary:
Algebraic injectivity was introduced to capture homotopical structures like algebraic Kan complexes. But at a much simpler level, it allows one to describe sets with operations subject to no equations. If one wishes to add equations (or operations of greater complexity) then it is natural to consider iterated algebraic injectives, which we introduce and study in the present paper. Our main application concerns Grothendieck’s weak $\omega$-groupoids, introduced in Pursuing Stacks, and the closely related definition of weak $\omega$-category due to Maltsiniotis. Using $\omega$ iterations we describe these as iterated algebraic injectives and, via this correspondence, prove the faithfulness conjecture of Maltsiniotis. Through work of Ara, this implies a tight correspondence between the weak $\omega$-categories of Maltsiniotis and those of Batanin/Leinster.
References:
[1] Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Vol. 189 of London Mathematical Society Lecture Note Series. Cambridge University Press
[2] Ara, Dimitri: Sur les ∞-groupoïdes de Grothendieck et une variante ∞-categoríque. PhD thesis, Université Paris VII
[3] Ara, Dimitri: On the homotopy theory of Grothendieck ω-groupoids. Journal of Pure and Applied Algebra 217, 1237–1278 MR 3019735
[4] Batanin, Michael: Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics 136(1) 39–103 DOI 10.1006/aima.1998.1724
[5] Berger, Clemens: A Cellular Nerve for Higher Categories. Advances in Mathematics 169,(2002) 118–175 DOI 10.1006/aima.2001.2056 | MR 1916373
[6] Bird, Gregory: Limits in 2-categories of locally-presented categories. PhD thesis, University of Sydney
[7] Bourke, John: Grothendieck ω-groupoids as iterated injectives. Talk given at CT2016. Slides at http://mysite.science.uottawa.ca/phofstra/CT2016/slides/Bourke.pdf
[8] Bourke, John: Equipping weak equivalences with algebraic structure. Mathematische Zeitschrift 294(3), 995-1019 DOI 10.1007/s00209-019-02305-w | MR 4074030
[9] Bourke, J., Garner, R.: On semiflexible, flexible and pie algebras. Journal of Pure and Applied Algebra 217, 293–321 DOI 10.1016/j.jpaa.2012.06.002 | MR 2969254
[10] Bourke, J., Garner, R.: Algebraic weak factorisation systems II: Categories of weak maps. Journal of Pure and Applied Algebra 220, 148–174 DOI 10.1016/j.jpaa.2015.06.003 | MR 3393454
[11] Bourke, J, Garner, R.: Monads and theories. Advances in Mathematics 251, 1024–1071 MR 3956765
[12] Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien, vol. 221 of Lecture Notes in Mathematics. Springer-Verlag
[13] Garner, Richard: Understanding the small object argument. Applied Categorical Structures 17(3), 247–285 DOI 10.1007/s10485-008-9137-4 | MR 2506256
[14] Grothendieck, Alexander: Pursuing Stacks.
[15] Henry, Simon: Algebraic models of homotopy types and the homotopy hypothesis. Arxiv Preprint 2016, https://arxiv.org/abs/1609.04622
[16] Lack, Steve: Homotopy-theoretic aspects of 2-monads. Journal of Homotopy and Related Structures 7(2), 229–260 MR 2369168
[17] Lanari, Edoardo: Towards a globular path object for weak ∞-groupoids. Journal of Pure and Applied Algebra 224, 630–702 DOI 10.1016/j.jpaa.2019.06.004 | MR 3987970
[18] Leinster, Tom: A survey of definitions of n-category. Theory and Applications of Categories 10 1–70 DOI 10.70930/tac/683ufulm | MR 1883478
[19] Maltsiniotis, Georges: Grothendieck ω-groupoids, and still another definition of ω-categories. Arxiv Preprint 2010, https://arxiv.org/abs/1009.2331
[20] Nikolaus, Thomas: Algebraic models for higher categories. Indag. Math. (N.S.) 21, no. 1–2, 52–75 DOI 10.1016/j.indag.2010.12.004 | MR 2832482
[21] Street, Ross: The petit topos of globular sets. Journal of Pure and Applied Algebra 154 299–315 DOI 10.1016/S0022-4049(99)00183-8
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