Yoneda lemma and representation theorem for double categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912088937463808 |
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| author | Fröhlich, Benedikt Moser, Lyne |
| author_facet | Fröhlich, Benedikt Moser, Lyne |
| contents | We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for the theory of double categories they play a similar role as 2-functors valued in $\mathrm{Cat}$ for 2-categories. We first introduce representable lax double presheaves and establish a Yoneda lemma. Then we build a Grothendieck construction which gives a 2-equivalence between lax double presheaves and discrete double fibrations over a fixed double category. Finally, we prove a representation theorem showing that a lax double presheaf is represented by an object if and only if its Grothendieck construction has a double terminal object. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Yoneda lemma and representation theorem for double categories Fröhlich, Benedikt Moser, Lyne Category Theory 18N10, 18D30, 18D60, 18D40 We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for the theory of double categories they play a similar role as 2-functors valued in $\mathrm{Cat}$ for 2-categories. We first introduce representable lax double presheaves and establish a Yoneda lemma. Then we build a Grothendieck construction which gives a 2-equivalence between lax double presheaves and discrete double fibrations over a fixed double category. Finally, we prove a representation theorem showing that a lax double presheaf is represented by an object if and only if its Grothendieck construction has a double terminal object. |
| title | Yoneda lemma and representation theorem for double categories |
| topic | Category Theory 18N10, 18D30, 18D60, 18D40 |
| url | https://arxiv.org/abs/2402.10640 |