Displayed Categories

Ahrens, Benedikt; Lumsdaine, Peter LeFanu

doi:10.23638/lmcs-15(1:20)2019

Cited by 11 publications

(17 citation statements)

References 3 publications

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“…The second available strategy has revealed practicable in a more efficient way: we define the desired category in a step-by-step construction by adding layers to a base category already given. Such a notion of layer corresponds precisely to a displayed category [6]: displayed categories are type-theoretic counterpart of fibrations and constitute a widely adopted instrument to reason about categories even at higher dimensions [2] in the UniMath library.…”

Section: Categorical Structuresmentioning

confidence: 99%

“…Having signatures, we then have given the related formalization of the category of algebras using the notion of a displayed category [6] over the category of sorted hSets, whose univalence is proven by adapting the strategy used for the univalence of functor categories. The resulting construction is still a modular one and the resulting proof-term is more concise, for sure, than the one obtained by checking that algebras and homomorphisms satisfy the axioms for standard categories.…”

Section: Introductionmentioning

confidence: 99%

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Universal Algebra in UniMath

Amato¹,

Maggesi²,

Brogi³

2021

Preprint

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We present recent updates in our development of a library for Universal Algebra in the UniMath proof assistant. The code here discussed concerns multi-sorted signatures and their algebras, along with the basics for equation systems. Moreover, we give neat constructions of the corresponding univalent categories by using the formalism of displayed categories, and show that the term algebra over a signature is the initial object of the category.Besides the formalization, we reflect on the methodological principles -based on the idea of evaluability of our elementary construction by the built-in normalization procedure of the systemleading our coding style, and show that this path is practicable indeed by sketching simple examples.

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Section: Categorical Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Universal Algebra in UniMath

Amato¹,

Maggesi²,

Brogi³

2021

Preprint

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“…In this work, we develop the notion of displayed bicategory analogous to the 1-categorical notion of displayed category introduced in [3]. Intuitively, a displayed bicategory D over a bicategory B represents data and properties to be added to B to form a new bicategory: D gives rise to the total bicategory D. Its cells are pairs (b, d) where d in D is a "displayed cell" over b in B. Univalence of D can be shown from univalence of B and "displayed univalence" of D. The latter two conditions are easier to show, sometimes significantly easier.…”

Section: Displayed Bicategoriesmentioning

confidence: 99%

“…Our work extends the notion of univalence from 1-categories [2] to bicategories. Similarly, we extend the notion of displayed 1-category [3] to the bicategorical setting. Capriotti and Kraus [7] study univalent (n, 1)-categories for n ∈ {0, 1, 2}.…”

Section: Related Workmentioning

confidence: 99%

“…Specifically, we give a notion of a univalent bicategory and show that some bicategories of interest are univalent, with examples from algebra and type theory. To this end, we generalize displayed categories of Ahrens and Lumsdaine [3] to the bicategorical setting, and prove that the total bicategory spanned by a displayed bicategory is univalent, if the constituent pieces are. In addition, we show how to embed bicategories of which the hom-categories are univalent into univalent ones via the Yoneda lemma, and we show how to use displayed machinery to construct biequivalences between total bicategories.…”

Section: Introductionmentioning

confidence: 99%

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Bicategories in Univalent Foundations

Ahrens,

Frumin,

Maggesi

et al. 2019

Preprint

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We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories.To construct examples of univalent bicategories, we develop the notion of displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. Displayed bicategories allow us to construct univalent bicategories in a modular fashion. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-category is weakly equivalent to a univalent bicategory.All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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What should a generic object be?

Sterling

2023

Math. Struct. Comp. Sci.

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Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $\lambda2$ -fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out some fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in this sense. We propose a new alignment of terminology that emphasizes the forms of generic object appearing most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphism. In addition, we propose a new class of acyclic generic objects inspired by recent developments in higher category theory and the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration.

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Displayed Categories

Cited by 11 publications

References 3 publications

Universal Algebra in UniMath

Universal Algebra in UniMath

Bicategories in Univalent Foundations

What should a generic object be?

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