| Title:
|
Non-unital polygraphs form a presheaf category (English) |
| Author:
|
Henry, Simon |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
3 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
248-291 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for proving that certain classes of polygraphs are presheaf categories. This criterion also applies to the larger class of polygraphs where only the source of each generator is not an identity, and to the class of "many-to-one polygraphs", producing a new, more direct, proof that this is a presheaf category. The criterion itself seems to be extendable to more general type of operads with possibly different combinatorics, but we leave this question for future work. In an appendix we explain why this result is relevant if one wants to fix the arguments of a famous paper of M.Kapranov and V.Voevodsky and make them into a proof of C. Simpson’s semi-strictification conjecture. We present a program aiming at proving this conjecture, which will be continued in subsequent papers. (English) |
| Keyword:
|
Polygraphs |
| Keyword:
|
Simpson’s semi-strictification conjecture |
| MSC:
|
18D05 |
| idZBL:
|
Zbl 1427.18014 |
| idMR:
|
MR3939049 |
| DOI:
|
10.21136/HS.2019.06 |
| . |
| Date available:
|
2026-03-10T23:27:01Z |
| Last updated:
|
2026-03-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153413 |
| . |
| Reference:
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[1] Batanin, Michael A.: Computads for finitary monads on globular sets..Contemporary Mathematics, 230:37–58 |
| Reference:
|
[2] Batanin, Michael A.: Computads and slices of operads..ArXiv preprint math/0209035 |
| Reference:
|
[3] Burroni, Albert: Higher-dimensional word problems with applications to equational logic..Theoretical computer science, 115(1):43–62 |
| Reference:
|
[4] Burroni, Albert: Automates et grammaires polygraphiques..Diagrammes, 67(68):9–32 |
| Reference:
|
[5] Carboni, Aurelio, Johnstone, Peter: Connected limits, familial representability and artin glueing..Mathematical Structures in Computer Science, 5(4):441–459 |
| Reference:
|
[6] Carboni, Aurelio, Johnstone, Peter: Corrigenda for ‘connected limits, familial representability and artin glueing’..Mathematical Structures in Computer Science, 14(1):185–187 |
| Reference:
|
[7] Cheng, Eugenia: A direct proof that the category of 3-computads is not cartesian closed..Arxiv:1209.0414 http://arxiv.org/pdf/1209.0414 |
| Reference:
|
[8] Forest, Simon, Mimram, Samuel: Unifying notions of pasting diagrams..to appear |
| Reference:
|
[9] Hadzihasanovic, Amar: A combinatorial-topological shape category for polygraphs..Arxiv:1806.10353 http://arxiv.org/pdf/1806.10353 |
| Reference:
|
[10] Harnik, Victor, Makkai, Michael, Zawadowski, Marek: Computads and multitopic sets..Arxiv:0811.3215 http://arxiv.org/pdf/0811.3215 |
| Reference:
|
[11] Henry, Simon: Regular polygraphs and the Simpson conjecture..Arxiv:1807.02627 http://arxiv.org/pdf/1807.02627 |
| Reference:
|
[12] Henry, Simon: Weak model categories in constructive and classical mathematics..Arxiv:1807.02650 http://arxiv.org/pdf/1807.02650 |
| Reference:
|
[13] Hermida, Claudio, Makkai, Michael, Power, John: On weak higher dimensional categories I..Journal of pure and applied algebra, 154(1):221–246 |
| Reference:
|
[14] Johnson, Michael: The combinatorics of n-categorical pasting..Journal of Pure and Applied Algebra, 62(3):211–225 |
| Reference:
|
[15] Joyal, André, Kock, Joachim: Weak units and homotopy 3-types..ArXiv preprint math/0602084 |
| Reference:
|
[16] Kapranov, Mikhail M, Voevodsky, Vladimir A.: Combinatorial-geometric aspects of polycategory theory: pasting chemes and higher bruhat orders (list of results)..Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32(1):11–27 |
| Reference:
|
[17] Kapranov, Mikhail M, Voevodsky, Vladimir A.: ∞-groupoids and homotopy types..Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32(1):29–46 |
| Reference:
|
[18] Kock, Joachim: Weak identity arrows in higher categories..International Mathematics Research Papers |
| Reference:
|
[19] Leinster, Tom: Higher operads, higher categories, volume 298..Cambridge University Press |
| Reference:
|
[20] Makkai, Michael: The word problem for computads..Available on the author’s web page |
| Reference:
|
[21] Makkai, Mihaly, Zawadowski, Marek: The category of 3-computads is not cartesian closed..Journal of Pure and Applied Algebra, 212(11):2543–2546 |
| Reference:
|
[22] Métayer, François: Cofibrant objects among higher-dimensional categories..Homology, Homotopy & Applications, 10(1) |
| Reference:
|
[23] Métayer, François: Strict ω-categories are monadic over polygraphs..Theory and Applications of Categories, 31(27):799–806 |
| Reference:
|
[24] Penon, Jacques: Approche polygraphique des ∞-categories non strictes..Cahiers de Topologie et Géométrie Différentielle, 40(1):31–80 |
| Reference:
|
[25] Power, A John: An n-categorical pasting theorem..In Category Theory, Springer lecture notes in Mathematics 1488, pages 326–358. Springer |
| Reference:
|
[26] Simpson, Carlos: Homotopy types of strict 3-groupoids..ArXiv preprint math/9810059 |
| Reference:
|
[27] Steiner, Richard: Omega-categories and chain complexes..Homology, Homotopy and Applications, 6(1):175–200 |
| Reference:
|
[28] Street, Ross: Limits indexed by category-valued 2-functors..Journal of Pure and Applied Algebra, 8(2):149–181 |
| Reference:
|
[29] Street, Ross: The algebra of oriented simplexes..Journal of Pure and Applied Algebra, 49(3):283–335 |
| Reference:
|
[30] Street, Ross: Parity complexes..Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32(4):315–343 |
| Reference:
|
[31] Thanh, Cédric Ho: The equivalence between many-to-one polygraphs and opetopic sets..Arxiv:1806.08645 http://arxiv.org/pdf/1806.08645 |
| . |
