| Title:
|
Operadic categories and their skew monoidal categories of collections (English) |
| Author:
|
Lack, Stephen |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
2 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
1-29 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category $\mathcal{S}$ of finite sets and functions, for which an operad is just a symmetric operad in the usual sense, the first skew monoidal category has underlying category [$\Bbb N, \bf Set$], and the second is the usual monoidal category of collections [$\Bbb P, \bf Set$] with the substitution monoidal structure. (English) |
| Keyword:
|
operad |
| Keyword:
|
skew monoidal category |
| Keyword:
|
operadic category |
| MSC:
|
18D10 |
| MSC:
|
18D50 |
| idZBL:
|
Zbl 1410.18012 |
| idMR:
|
MR3917424 |
| DOI:
|
10.21136/HS.2018.01 |
| . |
| Date available:
|
2026-03-10T14:04:17Z |
| Last updated:
|
2026-03-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153400 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Batanin, M. A.: Monoidal globular categories as a natural environment for the theory of weak n-categories..Adv. Math., 136(1):39–103 |
| Reference:
|
[4] Batanin, Michael, Markl, Martin: Operadic categories and duoidal Deligne’s conjecture..Adv. Math., 285:1630–1687 |
| Reference:
|
[5] Boardman, J. M., Vogt, R. M.: Homotopy invariant algebraic structures on topological spaces..Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York |
| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[9] Lack, Stephen, Street, Ross: Skew monoidales, skew warpings and quantum categories..Theory Appl. Categ., 26:385–402 |
| Reference:
|
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| Reference:
|
[11] Leinster, Tom: Higher operads, higher categories, volume 298 of LMS Lecture Note Series..Cambridge University Press, Cambridge |
| Reference:
|
[12] Loday, Jean-Louis, Stasheff, James, Voronov, Alexander: Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Mathematics..AMS |
| Reference:
|
[13] May, J. P.: The geometry of iterated loop spaces..Springer-Verlag, Berlin. Lectures Notes in Mathematics, Vol. 271 |
| Reference:
|
[14] Szlachányi, Kornél: Skew-monoidal categories and bialgebroids..Adv. Math., 231(3-4):1694–1730 |
| . |
