A note on relations relative to a factorization system

1991 ◽  
pp. 249-261 ◽  
Author(s):  
G. M. Kelly
Keyword(s):  
Factorization System

On some orthogonal factorization systems

2015 ◽  
Vol 14 (08) ◽  
pp. 1550120
Author(s):  
Othman Echi ◽  
Sami Lazaar ◽  
Mohamed Oueld Abdallahi
Keyword(s):  
Galois Connection ◽  
Orthogonality Relation ◽  
Topological Spaces ◽  
Orthogonal Factorization ◽  
Factorization System ◽  
Orthogonal Factorization System

The orthogonality relation between arrows in the class of all morphisms of a given category C yields a "concrete" antitone Galois connection between the class of all subclasses of morphisms of C. For a class Σ of morphisms of C, we denote by ⊥Σ (resp., Σ⊥) the class of all morphisms f in C such that f ⊥ g (resp., g ⊥ f) for each morphism g in Σ. A couple (Σ, Γ) of classes of morphisms is said to be an (orthogonal) prefactorization system if If, in addition the pfs satisfies then it will be called a dense prefactorization system. A pair [Formula: see text] of classes of morphisms in a category C is called an (orthogonal) factorization system if it is a prefactorization system and each morphism f in C has a factorization f = me, with [Formula: see text] and [Formula: see text]. This paper provides several examples of factorization systems and dense factorization systems in the category Top of topological spaces.

scholarly journals Modal descent

2020 ◽  
pp. 1-29
Author(s):  
Felix Cherubini ◽  
Egbert Rijke
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Orthogonal Factorization ◽  
Simple Description ◽  
Factorization System ◽  
Homotopy Type Theory ◽  
Orthogonal Factorization System ◽  
The Right ◽  
Special Case

Abstract Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$ . We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

scholarly journals Some categorical aspects of BCH-algebras

2003 ◽  
Vol 2003 (27) ◽  
pp. 1739-1750
Author(s):  
Muhammad Anwar Chaudhry ◽  
Hafiz Fakhar-ud-din
Keyword(s):  
Open Problem ◽  
Factorization System ◽  
One To One

We show that the category BCH of BCH-algebras and BCH-homomorphisms is complete. We also show that it has coequalizers, kernel pairs, and an image factorization system. It is also proved that onto homomorphisms and coequalizers, and monomorphisms and one-to-one homomorphisms coincide, respectively, in BCH. It is shown that MBCI is a coreflexive subcategory of BCH. Regular homomorphisms have been defined and their properties are studied. An open problem has been posed.

scholarly journals On factorization systems for surjective quandle homomorphisms

2014 ◽  
Vol 23 (11) ◽  
pp. 1450060 ◽  
Author(s):  
Valérian Even ◽  
Marino Gran
Keyword(s):  
Special Class ◽  
Precise Description ◽  
Factorization System ◽  
The One ◽  
Composition Of Relations ◽  
The Way

We study and compare two factorization systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorization system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorization system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorization systems.

scholarly journals A Relative Monotone-Light Factorization System for Internal Groupoids

2018 ◽  
Vol 26 (5) ◽  
pp. 931-942 ◽  
Author(s):  
Alan S. Cigoli ◽  
Tomas Everaert ◽  
Marino Gran
Keyword(s):  
Factorization System ◽  
Internal Groupoids

scholarly journals Identity types and weak factorization systems in Cauchy complete categories

2019 ◽  
Vol 29 (9) ◽  
pp. 1411-1427
Author(s):  
Paige Randall North
Keyword(s):  
Type Theory ◽  
Search Space ◽  
Proper Class ◽  
Factorization System ◽  
Dependent Type Theory ◽  
Weak Factorization System ◽  
Weak Factorization ◽  
Original Question ◽  
Dependent Type

AbstractIt has been known that categorical interpretations of dependent type theory with Σ- and Id-types induce weak factorization systems. When one has a weak factorization system $({\cal L},{\cal R})$ on a category $\mathbb{C}$ in hand, it is then natural to ask whether or not $({\cal L},{\cal R})$ harbors an interpretation of dependent type theory with Σ- and Id- (and possibly Π-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class ${\cal D}$ of morphisms of $\mathbb{C}$ such that the retract closure of ${\cal D}$ is the class ${\cal R}$ and the pair $(\mathbb{C},{\cal D})$ forms a display map category modeling Σ- and Id- (and possibly Π-) types. In this paper, we show, with the hypothesis that $\cal{C}$ is Cauchy complete, that there exists such a class $\cal{D}$ if and only if $(\mathbb{C},\cal{R})$itself forms a display map category modeling Σ- and Id- (and possibly Π-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.

Logic, sheaves, and factorization systems

1993 ◽  
Vol 58 (3) ◽  
pp. 872-893
Author(s):  
G. P. Monro
Keyword(s):  
Full Subcategory ◽  
Functional Relation ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
Completeness Proof ◽  
Factorization System ◽  
First Order ◽  
Functional Relations

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

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