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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1994 • Question 14

Question

Math 6310 • Algebra Fall 1994 • Question 14

  1. Explain what it means for a contravariant functor \(F\colon \mathcal {C}\to \mathbf {Set}\) to be representable (‘Set’ is the category of sets).
  2. Show how the power set \(P(S)=\{T\mid T\subseteq S\}\) defines a contravariant functor \(P\colon \mathbf {Set}\to \mathbf {Set}\) and prove that \(P\) is representable.

Raw LaTeX

\begin{enumerate}
\item[(a)] Explain what it means for a contravariant functor
\(F\colon \mathcal{C}\to \mathbf{Set}\) to be representable (`Set' is the category of sets).
\item[(b)] Show how the power set \(P(S)=\{T\mid T\subseteq S\}\) defines a contravariant functor
\(P\colon \mathbf{Set}\to \mathbf{Set}\) and prove that \(P\) is representable.
\end{enumerate}

Topic Tags

Ext Functors and Module Extensions Hom Functors and Module Homomorphisms Tor Functors and Projective Resolutions