Essentially surjective functor

An essentially surjective functor is a term from the mathematical branch of category theory .

definition

A functor between two categories and is called essentially surjective (or dense ) if for every object in an object in exists such that it is isomorphic to . ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle D}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F(C)}$ ${\displaystyle D}$

Examples

• Every equivalence of categories yields an essentially surjective functor, because a functor is an equivalence if and only if it is completely true and essentially surjective.
• Conversely, essential surjectivity can also be characterized by equivalence: a functor is then essentially surjective if and only if the full sub-category of equivalent to is generated by the image of the objects in . ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$
• If a field is the category of the vector spaces (in the sense of the -fold direct sum ), the cardinal number , and the category of all -vector spaces, then the embedding is essentially surjective, because according to the results of linear algebra every vector space is isomorphic to one . ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle K^{n}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle K}$ ${\displaystyle K^{n}}$
• If the body of the real or complex numbers , the category of Hilbert spaces over the isometric isomorphism and the category of the quantities with the bijective mappings from which is according to the set of Fischer-Riesz the functor substantially onto. ${\displaystyle K}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \ell ^{2}\colon {\mathcal {C}}\to {\mathcal {D}}}$

Individual evidence

1. ↑ Gerd Laures, Markus Szymik: Basic Topology Course , Spectrum Academic Publishing House 2009, ISBN 3827420407 , page 130
2. ↑ Gerd Laures, Markus Szymik: Basic Topology Course , Spectrum Academic Publishing House 2009, ISBN 3827420407 , sentence 7.5