# Fibrations of predicates and bicategories of relations

@article{Lawler2014FibrationsOP, title={Fibrations of predicates and bicategories of relations}, author={Finn Lawler}, journal={arXiv: Category Theory}, year={2014} }

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…

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## References

SHOWING 1-10 OF 117 REFERENCES

Framed bicategories and monoidal fibrations

- Mathematics
- 2007

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors,…

Maps II: Chasing Diagrams in Categorical Proof Theory

- Mathematics, Computer ScienceLog. J. IGPL
- 1996

This paper investigates proof theory of regular logic the {A,3}-fragment of the first order logic with equality, and determines precise conditions under which a regular fibration supports the principle of function comprehension, thus lifting a basic theorem from regular categories.

A $2$-categorical approach to change of base and geometric morphisms I

- Mathematics
- 1991

We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK ,a nd proK for a suitable category K, along…

Coherent extensions and relational algebras

- Mathematics
- 1974

ABSTRACT. The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is…

Enhanced 2-categories and limits for lax morphisms

- Mathematics
- 2012

Abstract We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or…

ON PROPERTY-LIKE STRUCTURES

- Mathematics
- 1997

A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2-category…

Limits indexed by category-valued 2-functors

- Mathematics
- 1976

There is common agreement now on the correct general notion of limit for categories whose horns are enriched in a suitable category 13. The definition involves a v-functor J: A + v which should be…

Enriched categories as a free cocompletion

- Mathematics
- 2013

Abstract This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory—categorifying the classical theory of categories enriched in a monoidal…

All realizability is relative

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2006

We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a pre-realizability notion. To each such object we can…

The formal theory of monads II

- Mathematics
- 2002

Abstract We give an explicit description of the free completion EM ( K ) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the…