Also, we give some examples and results on induced crossed modules in the case when 𝜙 is an epimorphism or the inclusion of an ideal.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper we examine on a pair of adjoint functors (𝜙∗ , 𝜙∗)for a subcategory of the category of crossed modules over commutative algebras where 𝜙∗: XMod/𝑃 → XMod/ Q, induced, and 𝜙∗: XMod/Q → XMod/𝑃, pullback (co-induced), which enables us to move from crossed 𝑄-modules to crossed P-modules by an algebra morphism 𝜙 : P → Q.
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Key Findings & Data:We show that this adjoint functor pair (𝜙∗, 𝜙∗) makes 𝑝 ∶ XMod → k-Alg into a bi- fibred category over k-Alg, the category of commutative algebras, where p is given by 𝑝(𝐶, 𝑅, 𝜕) = 𝑅.
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Conclusion & Impact:Also, we give some examples and results on induced crossed modules in the case when 𝜙 is an epimorphism or the inclusion of an ideal.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this paper we examine on a pair of adjoint functors (𝜙∗ , 𝜙∗)for a subcategory of the category of crossed modules over commutative algebras where 𝜙∗: XMod/𝑃 → XMod/ Q, induced, and 𝜙∗: XMod/Q → XMod/𝑃, pullback (co-induced), which enables us to move from crossed 𝑄-modules to crossed P-modules by an algebra morphism 𝜙 : P → Q. We show that this adjoint functor pair (𝜙∗, 𝜙∗) makes 𝑝 ∶ XMod → k-Alg into a bi- fibred category over k-Alg, the category of commutative algebras, where p is given by 𝑝(𝐶, 𝑅, 𝜕) = 𝑅. Also, we give some examples and results on induced crossed modules in the case when 𝜙 is an epimorphism or the inclusion of an ideal.
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Cite This Article
ALANSAL, Ö. G., ARSLAN, U. E. (2022). Crossed modules bifibred over k-Algebras. Cumhuriyet Science Journal, Cilt: 42 - Sayı: 1, 99–114. https://ajindex.com/articles/477614
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