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\title{Problem Set 5: Category theory}
\date{Due: Thursday, March 10}
\author{Your name:}
\begin{document}
\maketitle
Read sections 1.1--1.4 and 3.1 of \emph{Categories in Context} by Emily Riehl (link \href{http://www.math.jhu.edu/~eriehl/context.pdf}{here} and on course webpage) OR read sections 1.4, 2.4, 3.1, 3.3, 3.4, 5.4 of \emph{Category theory} by Steve Awodey (link \href{http://www.mpi-sws.org/~dreyer/courses/catlogic/awodey.pdf}{here} and on course webpage).
\begin{prob}[8]
Did you do your reading assignment?
\end{prob}
\begin{prob}[8] \noindent
\begin{enumerate}
\item What is a functor between groups, regarded as one-object categories? What is a natural transformation between a parallel pair of such functors?
\item What is a functor between preorders, regarded as categories? What is a natural transformation between a parallel pair of such functors?
\end{enumerate}
\end{prob}
\begin{prob}[8]
Prove that functors carry commutative diagrams to commutative diagrams. Note: part of this exercise is to formalize the notion of commutative diagram.
\end{prob}
\begin{prob}[8]
Let $X$ be a space and $L_X$ be the category associated to the directed set $(\t_X, \subset)$. Convince yourself that you know what the objects and morphisms of $L_X$ are.
\begin{enumerate}
\item In what way is the construction $X \mapsto L_X$ functorial? (That is, how can you make it into a functor?)
\item Let $\{X_{i}\}_{i \in I}$ be a family of objects of $L_X$. Does the product $\underset{i \in I} {\prod} X_i$ exist? Does the coproduct $\underset{i \in I} {\coprod} X_i$ exist? When?
\end{enumerate}
\end{prob}
\begin{prob}[16]
Let $C$ be a category. For a diagram $X: J \to C$, denote the colimit of the diagram (if it exists) by $(\underset{J}{\colim}(X), \{ \iota_j \}_{j \in \ob J} )$ where $$\underset{J}{\colim}(X) \in \ob C$$ is the universal object of the colimit and $$\iota_{j}: X(j) \to \underset{J}{\colim}(X)$$ are the universal morphisms.
A functor $F: C \to D$ is said to \emph{preserve colimits} if for every diagram $X: J \to C$ such that the colimit of $X$ exists, the colimit $\underset{J}{\colim}(FX)$ also exists, and the map $$\underset{J}{\colim}(FX) \to F \left( \underset{J}{\colim}{X} \right),$$ which is induced by the maps $F \iota_{j} : F(X(j)) \to F (\underset{J}{\colim}(X) )$ and the universal property of $\underset{J}{\colim}(FX),$ is an isomorphism.
\begin{enumerate}
\item Give an example of a functor that \emph{does not} preserve colimits.
\item Give an example of a functor that preserves colimits.
\item Let $C$ be a category. Prove that any functor $F: \text{Set} \to C$ that preserves colimits is determined by the image under $F$ of any set with one-element.
\item Let $C$ be a category. Let $D$ be one of the following categories: $\text{Group}, \text{Vect}_{\R}, \text{Top}$. Is there a collection of subobjects $G \subset \ob D$ so that any functor $F: D \to C$ that preserves colimits is determined by its values on $G$? What if we allow $G$ to be a subcategory?
\end{enumerate}
\end{prob}
\end{document}