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- W4287668766 abstract "We introduce a functor $mathcal Vcolon mathrm{DblCat}_{h,nps}to mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of bi-representations of a normal pseudo-functor $Fcolon mathbf C^{operatorname{op}}to mathrm{Cat}$ in terms of double bi-initial objects in the double category $mathbb{E}l(F)$ of elements of $F$, or equivalently as bi-initial objects of a special form in the $2$-category $mathcal Vmathbb{E}l(F)$ of morphisms of $F$. Although not true in general, in the special case where the $2$-category $mathbf C$ has tensors by the category $mathbf{2}={0to 1}$ and $F$ preserves those tensors, we show that a bi-representation of $F$ is then precisely a bi-initial object in the $2$-category $mathbf{E}l(F)$ of elements of $F$. We give applications of this theory to bi-adjunctions and weighted bi-limits." @default.
- W4287668766 created "2022-07-25" @default.
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- W4287668766 date "2020-09-11" @default.
- W4287668766 modified "2023-09-23" @default.
- W4287668766 title "Bi-initial objects and bi-representations are not so different" @default.
- W4287668766 doi "https://doi.org/10.48550/arxiv.2009.05545" @default.
- W4287668766 hasPublicationYear "2020" @default.
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