In this thesis, we develop the theory of bifibrations of polycategories. We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon representability and look at different variations, namely the correspondence between representable multicategories and monoidal categories, birepresentable polycategories and $\ast$-autonomous categories, and representable virtual double categories and double categories. We then move to introduce (bi)fibrations for these structures. We show that it generalises representability in the sense that these structures are (bi)representable when they are (bi)fibred over the terminal one. We show how to use this theory to lift models of logic to more refined ones. In particular, we illustrate it by lifting the compact closed structure of the category of finite dimensional vector spaces and linear maps to the (non-compact) $\ast$-autonomous structure of the category of finite dimensional Banach spaces and contractive maps by passing to their respective polycategories. We also give an operational reading of this example, where polylinear maps correspond to operations between systems that can act on their inputs and whose outputs can be measured/probed and where norms correspond to properties of the systems that are preserved by the operations. Finally, we recall the B\'enabou-Grothendieck correspondence linking fibrations to indexed categories. We show how the B-G construction can be defined as a pullback of virtual double categories and we make use of fibrational properties of vdcs to get properties of this pullback. Then we provide a polycategorical version of the B-G correspondence.

翻译：本论文发展了多范畴的双纤维化理论。我们首先通过推广态射的形状，研究如何将某些范畴结构表达为泛性质，称此现象为可表性，并考察其不同变体：可表多范畴与幺半范畴、双可表多范畴与*−自伴范畴、可表虚双范畴与双范畴之间的对应关系。随后引入这些结构的（双）纤维化，证明这些结构在（双）纤维于终端结构时即为（双）可表，从而表明此理论推广了可表性。我们展示如何运用该理论将逻辑模型提升至更精细的模型。特别地，通过从有限维向量空间与线性映射范畴的紧*-自伴结构，经由相应多范畴的过渡，提升至有限维巴拿赫空间与压缩映射范畴的（非紧）*-自伴结构，以此阐释该理论。对此实例我们还给出操作性解读：多线性映射对应于可在输入上作用、输出可测量/探测的系统间操作，而范数则对应系统被操作所保持的性质。最后回顾将纤维化与索引范畴相关联的Bénabou-Grothendieck对应，证明B-G构造可定义为虚双范畴的回拉，并利用虚双范畴的纤维性质导出该回拉的性质，进而给出B-G对应的多范畴版本。