R. F. C. Walters, R. J. Wood
Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, map(B)(T,A) is a groupoid.
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Bicategories of spans are characterized as cartesian bicategories in which every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is comonadic.
The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory.
Relational structures are emerging as ubiquitous mathematical machinery in the semantics of open systems of various kinds. Cartesian bicategories are a well-known categorical algebra of relations that has proved especially useful in recent applications. The passage between a category and its bicategory of relations is an important question that has been widely studied for decades. We study an alternative construction that yields a cartesian bicategory of relations. Its behaviour is closely related to the axiom of choice, which itself can be expressed in the language of cartesian bicategories.
In this paper, we provide a notion of $\infty$-bicategories fibred in $\infty$-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen sca…
Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is the internal logic of regular categories. The link between these two facts is that in any regular category, one may construct a notion of binary relation using jointly-monic spans; this results in what is known as the bicategory of relations of the regular category. In this paper we provide a direct axiomatization of bicategories of relations, which we term …
We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the former in algebraic terms based on a string diagrammatic calculus, the latter in universal terms using the fundamental notion of adjoint functor. We prove that these two approaches are related by an adjunction, which can be strengthened to an equivalence by imposing further constraints on doctrines.
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the dif…
We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two are equivalent. A regular equipment is a `cartesian equipment' satisfying certain axioms, and a cartesian equipment is a slight generalization of a cartesian bicategory. This is done by defining a tricategory Biprof whose objects are bicategories and whose morphisms are category-valued profunctors, and then defining an equipment to be a pseudo-monad in this …
We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory.
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us to set the whole theory inside the welcoming 2-category of double categories, and to overcome technical problems that were caused by working with left adjoints inside a general bicategory. Cartesian double categories that are also fibrant are of particular interest to us. After describing some important properties of Cartesian and fibrant double categories, …
2-Cartesian fibrations II: A Grothendieck construction for $\infty$-bicategories
本文通过构造Lurie拉直-反拉直伴随的2-范畴版本，建立了缩放单纯集上2-笛卡尔纤维化与取值于∞-双范畴的反变函子之间的等价，并借助相对神经构造与已有双范畴格罗滕迪克构造进行比较。
A convenient model category for bicategories
本文引入并研究柔性2-范畴的模型范畴，它与Lack的2-范畴模型范畴Quillen等价，但具有所有对象都是余纤维的、关于笛卡尔闭结构是幺半模型范畴等优良性质，其纤维对象全子范畴等价于双范畴与正规伪函子的范畴。
Cartesian Fibrations of $(\infty,2)$-categories
本文在缩放单纯集框架下引入$\infty$-双范畴的四种笛卡尔2-纤维化变体，刻画其提升性质与相容条件，证明等价性可逐纤维检验，并建立富集$\infty$-范畴情形与缩放神经之间的对应关系。
Transposing cartesian and other structure in double categories
本文探讨了双范畴中笛卡尔结构与其他结构的转换，通过建立双范畴与双范畴的联系，证明了具有笛卡尔结构的双范畴下存在笛卡尔双范畴。
We introduce the concept of Frobenius theory as a generalisation of Lawvere's functorial semantics approach to categorical universal algebra. Whereas the universe for models of Lawvere theories is the category of sets and functions, or more generally cartesian categories, Frobenius theories take their models in the category of sets and relations, or more generally in cartesian bicategories.
Hilsum-Skandalis maps, from differential geometry, are studied in the context of a cartesian category. It is shown that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions. This leads to a new and more general proof that Hilsum-Skandalis maps represent a universal way of inverting essential equivalences between internal groupoids. To prove the representation theorem, a new characterisation of the con- nected components adjunction of any internal groupoid is given. The charaterisation is that the adjunction is covered by a stable Frobenius adjunction that is a slice and who…
From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness
本文扩展 gs-monoidal 范畴为带箭头预序的 oplax cartesian 范畴，刻画其 Kleisli 与 span 构造，并证明 Yoneda 嵌入及函子完备性定理。
Frobenius templicial modules and the dg-nerve
本文提出 Frobenius templicial 对象，证明正分次 dg-范畴与 Frobenius templicial 模之间的范畴等价，并由此给出经典 dg-神经的自然充实，刻画了模中拟范畴的 Frobenius 结构条件。
The Frobenius equivalence and Beck-Chevalley condition for Algebraic Weak Factorisation Systems
本文针对代数弱分解系统证明 Frobenius 等价与 Beck-Chevalley 条件的类比命题，指出其结构包含分解与提升，并基于类型论范畴语义动机，以群胚的分裂纤维化为例说明其应用。
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