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Exponential Objects

Marco Riccardi (2015)

Formalized Mathematics

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].

Extensional subobjects in categories of Ω -fuzzy sets

Jiří Močkoř (2007)

Czechoslovak Mathematical Journal

Two categories 𝕊𝕖𝕥 ( Ω ) and 𝕊𝕖𝕥𝔽 ( Ω ) of fuzzy sets over an M V -algebra Ω are investigated. Full subcategories of these categories are introduced consisting of objects ( s u b ( A , δ ) , σ ) , where s u b ( A , δ ) is a subset of all extensional subobjects of an object ( A , δ ) . It is proved that all these subcategories are quasi-reflective subcategories in the corresponding categories.

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