Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913605842108416 |
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| author | Arkor, Nathanael Bourke, John Ko, Joanna |
| author_facet | Arkor, Nathanael Bourke, John Ko, Joanna |
| contents | Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation Arkor, Nathanael Bourke, John Ko, Joanna Category Theory 18C10, 18C30, 18C40, 18D20, 18M65, 18N10 Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories. |
| title | Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation |
| topic | Category Theory 18C10, 18C30, 18C40, 18D20, 18M65, 18N10 |
| url | https://arxiv.org/abs/2412.07475 |