Strictification and non-strictification of monoidal categories
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929649667276800 |
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| author | Becerra, Jorge |
| author_facet | Becerra, Jorge |
| contents | In this survey paper we give account of several approaches to the strictification and non-strictification of monoidal categories, which are constructions that turn a monoidal category into a (non-)strict one monoidally equivalent to the original category, and how they are related to analogous notions in higher categorical structures. We first provide explicit, elementary models for the (non-)strictification and show that these two constructions give the free (non-)strict monoidal category generated by a monoidal category. Moreover, we prove in detail that these two constructions are part of a pair of free-forgetful 2-adjunctions. We later show that these constructions can be recovered from Power's general coherence theorem for 2-monads. Lastly we describe another model for the strictification based on right-module endofunctors and provide a detailed, self-contained proof that this is a particular instance of strictification of bicategories via the bicategorical analogue of the Yoneda embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strictification and non-strictification of monoidal categories Becerra, Jorge Category Theory Geometric Topology 18M05, 18N10, 18N15 In this survey paper we give account of several approaches to the strictification and non-strictification of monoidal categories, which are constructions that turn a monoidal category into a (non-)strict one monoidally equivalent to the original category, and how they are related to analogous notions in higher categorical structures. We first provide explicit, elementary models for the (non-)strictification and show that these two constructions give the free (non-)strict monoidal category generated by a monoidal category. Moreover, we prove in detail that these two constructions are part of a pair of free-forgetful 2-adjunctions. We later show that these constructions can be recovered from Power's general coherence theorem for 2-monads. Lastly we describe another model for the strictification based on right-module endofunctors and provide a detailed, self-contained proof that this is a particular instance of strictification of bicategories via the bicategorical analogue of the Yoneda embedding. |
| title | Strictification and non-strictification of monoidal categories |
| topic | Category Theory Geometric Topology 18M05, 18N10, 18N15 |
| url | https://arxiv.org/abs/2303.16740 |