Free object

Free objects are studied in abstract algebra . These are algebraic structures in which only those equations apply that follow from the defining axioms of the algebraic structure, that is, that are free from further relations. In category theory, free objects are defined by a universal property .

definition

Let it be a concrete category with the forget function . Optionally further be a lot , an object from and an injective map . The pair is called free over if the following universal property is fulfilled: ${\displaystyle (\mathrm {K} ,V)}$ ${\displaystyle V\colon \mathrm {K} \rightarrow \mathrm {Set} }$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle \mathrm {K} }$ ${\displaystyle i\colon X\rightarrow V(A)}$ ${\displaystyle (A,i)}$ ${\displaystyle X}$

For each object from and every picture there is exactly one morphism with , which means that the following diagram commutative is: ${\displaystyle B}$ ${\displaystyle \mathrm {K} }$ ${\displaystyle f\colon X\rightarrow V(B)}$ ${\displaystyle g\colon A\rightarrow B}$ ${\displaystyle f=V(g)\circ i}$

${\displaystyle {\begin{array}{c}X{\xrightarrow {\quad i\quad }}V(A)\\{}_{f}\searrow \quad \swarrow {}_{V(g)}\\V(B)\quad \\\end{array}}}$

Often and is the inclusion figure . Then you leave out and name, somewhat imprecisely, the free object . ${\displaystyle X\subset V(A)}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle A}$ ${\displaystyle X}$

Uniqueness

Are free over and free over and are and equal in power , then and are isomorphic . So if there are free objects, these are unique except for isomorphism and only depend on the thickness of the set. ${\displaystyle (A,i)}$ ${\displaystyle X}$ ${\displaystyle ({\tilde {A}},{\tilde {i}})}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle X}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$

Examples

The best known case is the category of vector spaces over a solid body with the K -linear mappings as morphisms. The forgetting functor maps a vector space onto the set of elements of the vector space, so it forgets the vector space structure. If there is a set, then there is an over free vector space. To do this, consider the vector space of all mappings with finite support . If the mapping that maps to 1 and every other element from to 0 is an injective mapping and is free over in the sense of the above definition. is a base of . The uniqueness theorem is nothing more than the well-known theorem that vector spaces with bases of equal power are isomorphic. There is also the special feature that every vector space is free, because every vector space has a basis and is free above every basis. ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle f\colon X\rightarrow K}$ ${\displaystyle i_{x}\in E}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle i\colon X\rightarrow E,x\mapsto i_{x}}$ ${\displaystyle (E,i)}$ ${\displaystyle X}$ ${\displaystyle \{i_{x}\mid x\in X\}}$ ${\displaystyle E}$

Other examples are

• Free Abelian group
• Free Associative Algebra
• Free group
• Free Lie algebra
• Free magma
• Free module
• Free monoid

Freedom as a functor

The construction of the free object over a set assigns an object of the given category to each set, if free objects exist in the category , for example with images . If a figure is in the category , there is, by definition, exactly one morphism , so that , that is, the diagram ${\displaystyle X}$ ${\displaystyle \mathrm {K} }$ ${\displaystyle X\mapsto F(X)}$ ${\displaystyle i_{X}\colon X\rightarrow V(F(X))}$ ${\displaystyle \alpha \colon X\rightarrow Y}$ ${\displaystyle \mathrm {Set} }$ ${\displaystyle i_{Y}\circ \alpha \colon X\rightarrow V(F(Y))}$ ${\displaystyle g_{\alpha }\colon F(X)\rightarrow F(Y)}$ ${\displaystyle i_{Y}\circ \alpha =V(g_{\alpha })\circ i_{X}}$

${\displaystyle {\begin{array}{ccc}X&{\xrightarrow {\alpha }}&Y\\\downarrow _{i_{X}}&&\downarrow _{i_{Y}}\\V(F(X))&{\xrightarrow {V(g_{\alpha })}}&V(F(Y))\end{array}}}$

is commutative. If one sets , one obtains a functor which is left adjoint to the forget functor. Conversely, one can define freedom as a left adjoint functor to the forgetting functor. ${\displaystyle F(\alpha ):=g_{\alpha }}$ ${\displaystyle F\colon \mathrm {Set} \rightarrow \mathrm {K} }$

Individual evidence

1. ↑ Ulrich Knauer, Kolja Knauer: Discrete and Algebraic Structures - In Brief , Springer-Verlag (2015), ISBN 978-3-662-45176-2 , Chapter 11.4: Freedom
2. ↑ Thomas W. Hungerford: Algebra , Springer-Verlag (1974), ISBN 978-1-4612-6103-2 , Chapter I §7, Definition 7.7
3. ↑ Ulrich Knauer, Kolja Knauer: Discrete and Algebraic Structures - In Brief , Springer-Verlag (2015), ISBN 978-3-662-45176-2 , sentence 11.13
4. ↑ Thomas W. Hungerford: Algebra , Springer-Verlag (1974), ISBN 978-1-4612-6103-2 , Chapter I §7, Sentence 7.8
5. ^ PJ Hilton, Urs Stammbach: A Course in Homological Algebra , Springer-Verlag (1971), ISBN 978-0-387-90033-9 , Chapter II.10: Projective, Injective and Free Objects