Description:
Definitions from category theory
Content:
Definitions from category theory, in particular
for forms Form 188, Form 189, Form 190, and Form 192, Form 193 from Blass [1979]
Definition: Assume \(C\) is a category
- An object \(p\) in \(C\) is projective (in \(C\)) if
for all objects \(a\) and \(b\) in \(C\), if \(f\in \hom(a,b)\) is an
epimorphism and \(g\in \hom(p,b)\), then there is an \(h \in \hom(p,a)\)
such that \(f\circ h = g\).
- \(p\) is injective if for all objects \(a\) and
\(b\) in \(C\), if \(f\in\hom(b,a)\) is a monomorphism and \(g\in\hom(b,p)\)
then there is an \(h\in \hom(a,p)\) such that \(h\circ f=g\).
- An Abelian group \(G\) is projective if for all Abelian
groups \(A\) and \(B\) if \(f : A \rightarrow B\) is a homomorphism from \(A\)
onto \(B\) and \(g : G \rightarrow B\) is a homomorphism then there is a
homomorphism \(h : G \rightarrow A\) such that \(f\circ h = g\).
- An Abelian group \(G\) is injective if for all Abelian
groups \(A\) and \(B\), if \(f : B \rightarrow A\) is a one to one homomorphism
and \(g : B \rightarrow G\) is a homomorphism then there is a homomorphism
\(h : A \rightarrow G\) such that \(h\circ f = g\).
- A set \(X\) is projective if for all sets \(A\) and \(B\), for
all \(f\) from \(A\) onto \(B\) and for all \(g\) from \(X\) into \(B\) there is a
function \(h\) from \(X\) into \(A\) such that \(f\circ h = g\), (i.e., For
any \(X\)-indexed family \(\{ y_{t} : t\in X \}\) of non-empty sets there
is a choice function.)
Howard-Rubin number:
60
Type:
Definitions
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