Sub-functor

Sub-functions are defined in the mathematical sub-area of category theory. A set-valued functor is a sub- functor of another if there is a subset relationship between the images of the objects and an associated restriction relationship between the images of the morphisms .

definition

Let there be a category and two functors in the category of sets , both covariant or contravariant . is called a sub-function of , if ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle F, G \ colon {\ mathcal {C}} \ rightarrow {\ mathcal {Set}}}$ ${\ displaystyle G}$ ${\ displaystyle F}$

for all objects of true and ${\ displaystyle C}$ ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle G (C) \ subset F (C)}$
for all morphisms in is (or in the contravariant case). ${\ displaystyle f: C \ rightarrow D}$ ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle G (f) = F (f) | _ {G (C)}}$ ${\ displaystyle G (f) = F (f) | _ {G (D)}}$

The vertical line stands for the restriction of the figure to the specified amount.

In this case the notation is used . ${\ displaystyle G \ subset F}$

Examples

Let be the category of the groups and the functor that maps each group to the commutator group and each group homomorphism to the restriction to the commutator group. Since group homomorphisms map commutators back to commutators, you get a functor. After all, is the forget function . Then is . ${\ displaystyle {\ mathcal {Grp}}}$ ${\ displaystyle K: {\ mathcal {Grp}} \ rightarrow {\ mathcal {Grp}}}$ ${\ displaystyle G}$ ${\ displaystyle K (G): = \ langle [G, G] \ rangle}$ ${\ displaystyle V: {\ mathcal {Grp}} \ rightarrow {\ mathcal {Set}}}$ ${\ displaystyle V \ circ K \ subset V}$

The functors defined by sieves are exactly the sub-functors of the contravariant Hom functor .

Sub-functors of prawns

If there is a small category , then a functor from the dual category into the category of sets is called a preamble . The functor category of embossing with the natural transformations as morphisms is denoted by or . By definition, sub-objects of a preamble are equivalence classes of monomorphisms . One can show that each sub-object in is represented by a sub-functor. This means that there is also a sub-function in each of these equivalence classes. ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle {\ mathcal {C}} ^ {op} \ rightarrow {\ mathcal {Set}}}$ ${\ displaystyle {\ hat {\ mathcal {C}}}}$ ${\ displaystyle {\ mathcal {Set}} ^ {{\ mathcal {C}} ^ {op}}}$ ${\ displaystyle F}$ ${\ displaystyle G \ rightarrow F}$ ${\ displaystyle {\ hat {\ mathcal {C}}}}$

Individual evidence

↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , definition 3.4.12
^ H. Schubert : Categories II , Springer-Verlag (1970), ISBN 978-3-540-04866-4 , definition 20.1.2
↑ Oswald Wyler: Lecture Notes on Topoi and Quasitopoi , World Scientific Publishing (1991), ISBN 981-02-0153-2 , Chapter 26.4: Subfunctors classified by sieves
↑ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. I.4

[1] Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , definition 3.4.12

[2] H. Schubert : Categories II , Springer-Verlag (1970), ISBN 978-3-540-04866-4 , definition 20.1.2

[3] Oswald Wyler: Lecture Notes on Topoi and Quasitopoi , World Scientific Publishing (1991), ISBN 981-02-0153-2 , Chapter 26.4: Subfunctors classified by sieves

[4] Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. I.4