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faithful functor (Definition)

A functor $ T:\mathcal{C}\to\mathcal{D}$ is faithful if the arrow function of $ T$ is injective for every pair of objects in $ \mathcal{C}$. More precisely, for every pair $ C_1, C_2\in \operatorname{Ob}(\mathcal{C})$, the arrow function $ T_{(C_1,C_2)}$ of $ T:$

$\displaystyle T_{(C_1,C_2)}:\operatorname{hom_{\mathcal{C}}}(C_1,C_2)\to\operatorname{hom_{\mathcal{D}}}(T(C_1),T(C_2))$
given by $ T_{(C_1,C_2)}(f)=T(f)$ is an injection. In other words, $ T(f)=T(g)$ implies $ f=g$.



"faithful functor" is owned by CWoo.
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See Also: subcategory, full functor

Other names:  embedding functor
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Cross-references: implies, objects, injective, functor
There are 13 references to this entry.

This is version 5 of faithful functor, born on 2004-05-11, modified 2007-05-03.
Object id is 5849, canonical name is FaithfulFunctor.
Accessed 3331 times total.

Classification:
AMS MSC: 18A22 (Category theory; homological algebra :: General theory of categories and functors :: Special properties of functors )
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