202411121911

Tags : Category

Functors

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John Baez "Quantum Quandries: A Category-Theory Perspective"

… Every sufficiently good analogy is yearning to become a functor.

Definition

A Functor $F:C\to D$, between categories $C$ and $D$ consists of the following objects

• An object $Fc\in D$, for each $c\in C$.
• A morphism $Ff:Fc\to F{c}^{\prime }$ for each morphism $f:c\to c^{\prime }$, so that the domain and codomain of $Ff$ are equal to $F$ applied on domain and codomain of $f$ respectively These objects need to satisfy the following functoriality axioms
• For any composable pair $f,g$ in $C$, $Fg\cdot Ff=F(g\cdot f)$
• For each object $c$ in $C,F(1_c)=1_{F_c}$.

Important

Functors preserve isomorphism, but functors do not preserve epimorphisms and monomorphisms.

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References

Examples of Functors