| Title:
|
Operadic Fibrations and Unary Operadic 2-categories (English) |
| Author:
|
Trnka, Dominik |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
10 |
| Issue:
|
1 |
| Year:
|
2026 |
| Pages:
|
212-239 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce unary operadic 2-categories as a framework for an operadic Grothendieck construction of a categorical O-operad, O being a unary operadic category. The construction is a fully faithful functor $\int\odot$ which takes categorical O-operads to operadic functors over O, and we characterize its essential image by certain lifting properties. Such operadic functors are called operadic fibrations. Our theory is an extension of the discrete (unary) operadic case and, in some sense, of the classical Grothendieck construction of a categorical presheaf. For the terminal unary operadic category $\odot$, a categorical $\odot$-operad is a strict monoidal category V and its Grothendieck construction $\int \odot {\rm V}$ is connected to the `para' construction appearing in machine learning. The 2-categorical setting provides a characterization of O-operads valued in ${\rm V}$ as operadic functors ${\rm O} \rightarrow \int \odot {\rm V}$. Last, we describe a left adjoint to $\int\odot$. (English) |
| Keyword:
|
unary operadic category; 2-category; fibration; Grothendieck construction; monoidal category; operad; 2-categorical nerve; decalage |
| MSC:
|
18D30 |
| MSC:
|
18M05 |
| MSC:
|
18M60 |
| MSC:
|
18N10 |
| MSC:
|
18N50 |
| DOI:
|
10.21136/HS.2026.06 |
| . |
| Date available:
|
2026-09-17T12:46:51Z |
| Last updated:
|
2026-09-17 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153758 |
| . |
| Reference:
|
[1] Bashkirov, D.: Lattice operads and operad filtrations,.arxiv:2212.14833 http://arxiv.org/pdf/2212.14833 |
| Reference:
|
[2] Batanin, M.A.: The Eckmann–Hilton argument and higher operads,.Advances in Mathematics, 217(1):334–385 |
| Reference:
|
[3] Batanin, M.A., Markl, M.: Operadic categories and duoidal Deligne's conjecture..Advances in Mathematics, 285:1630-1687 |
| Reference:
|
[4] Batanin, M.A., Markl, M.: Operadic categories as a natural environment for Koszul duality,.Compositionality, 5(3) |
| Reference:
|
[5] Batanin, M.A., Markl, M.: Koszul duality for operadic categories,.Compositionality, 5(4) |
| Reference:
|
[6] Batanin, M.A., Markl, M.: Operads, operadic categories and the blob complex,.Applied Categorical Structures, 32(1) |
| Reference:
|
[7] Berger, C.: Combinatorial models for real configuration spaces and $E_(n)$-operads,.Contemporary Mathematics, 202 |
| Reference:
|
[8] Corner, A.S., Gurski, N.: Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat,.arxiv:1312.5910 http://arxiv.org/pdf/1312.5910 |
| Reference:
|
[9] Cruttwell, G.S.H., Gavranović, B., Ghani, N., Wilson, P., Zanasi, F.: Categorical foundations of gradient-based learning,.European Symposium on Programming |
| Reference:
|
[10] Duskin, J.: Simplicial matrices and the nerves of weak $n$-categories I: nerves of bicategories,.Theory and Applications of Categories, 9(10) |
| Reference:
|
[11] Fong, B., Spivak, D., Tuyéras, R.: Backprop as Functor: A compositional perspective on supervised learning,.Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, 11 |
| Reference:
|
[12] Fresse, B.: Homotopy of operads and Grothendieck-Teichmuller groups,.Mathematical surveys and monographs, 217 |
| Reference:
|
[13] Garner, R., Kock, J., Weber, M.: Operadic categories and décalage,.Advances in Mathematics, 377:107440 |
| Reference:
|
[14] Gavranović, B.: Compositional Deep Learning,.arxiv:1907.08292 http://arxiv.org/pdf/1907.08292 |
| Reference:
|
[15] Hackney, P.: Operadic categories and 2-Segal sets,.arxiv:2312.00756 http://arxiv.org/pdf/2312.00756 |
| Reference:
|
[16] Johnson, N., Yau, D.: 2-Dimensional Categories,.Oxford University Press |
| Reference:
|
[17] Lurie, J.: Kerodon,.https://kerodon.net |
| Reference:
|
[18] Markl, M.: Bivariant operadic categories,.arxiv:2402.12963 http://arxiv.org/pdf/2402.12963 |
| Reference:
|
[19] Morrison, S., Walker, K.: Blob homology,.Geometry & Topology, 16:1481–1607 |
| Reference:
|
[20] Štěpán, M.: Colax adjunctions and lax-idempotent pseudomonads,.Theory and Applications of Categories, 44(7):227-271 |
| Reference:
|
[21] Trnka, D.: Category-colored operads, internal operads, and Markl O-operads,.Theory and Applications of Categories, 39(30):874–915 |
| . |
