Functors map between categories preserving composition and identity
Image: Wassily Kandinsky, Public domain, via Wikimedia Commons
Functor
Functors map between categories preserving composition and identity
Functors are mappings between categories that maintain the structure of the categories they connect. They ensure that the composition of morphisms (functions) in one category corresponds to the composition of morphisms in another category. Additionally, functors preserve identity morphisms, meaning that the identity element in one category maps to the identity element in the target category.
Example
Consider two categories, Category A and Category B. A functor F maps objects from Category A to objects in Category B. If there are morphisms f and g in Category A such that f composed with g equals some morphism h in Category A, then F(f) composed with F(g) in Category B will equal F(h). Furthermore, if there is an identity morphism i in Category A, then F(i) will be the identity morphism in Category B.
Understanding functors is crucial in category theory as they provide a way to relate different mathematical structures while preserving their intrinsic properties. This concept is fundamental in modern mathematics and has applications across various fields.
Related concepts
Yoneda lemma
The Yoneda lemma embeds a locally small category into a functor category
the tokenizer's special tokens do: [CLS], [SEP], [PAD], [MASK] have specific roles
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Overlapping subproblems
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Inner product space
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