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\lecture{Subobjects}

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\maketitle

\begin{definition}[Monomorphism]
A morphism $\ob{C}_1 \xmto{\mo{m}} \ob{C}_2$ such that for all $\ob{C} \xtwomto[\mo{m}_2]{\mo{m}_1} \ob{C}_1$, if $\mo{m}_1 \cocomp \mo{m}$ equals $\mo{m}_2 \cocomp \mo{m}$ then $\mo{m}_1$ equals~$\mo{m}_2$.
\end{definition}

\begin{notation}
Monomorphisms are indicated by $\mono$.
\end{notation}

\begin{exercise}
Prove that the monomorphisms in~$\cat{Set}$, $\cat{Prost}$, $\mathbb{\Delta}_n$, $\mathbb{\Delta}$, and~$\cat{Mon}$ are the injective functions.
\end{exercise}

\begin{exercise}
Classify the monomorphims in~$\cat{Rel}$. Hint: while left-total left-unique relations (i.e.~relations corresponding to injective functions) are monomorphisms in~$\cat{Rel}$, not all monomorphisms in~$\cat{Rel}$ fall under that classification.
\end{exercise}

\begin{exercise}
Prove that monomorphisms in~$\cat{Mat}$ are the matrices where one can remove some select rows to obtain an invertible matrix.
\end{exercise}

\begin{exercise}
Prove that monomorphisms in~$\cat{CAT}$ are precisely the functors whose component on objects and components on morphisms are all injective functions.
\end{exercise}

\begin{exercise}
Prove that all sections are monomorphisms, but not all monomorphisms are sections. Hint: use~$\cat{Prost}$ for your counterexample.
\end{exercise}

\begin{exercise}
Prove that all retracts that are monomorphisms are also isomorphisms.
\end{exercise}

\begin{remark}
The above generalizes the proof that a surjection that is injective is a bijection.
\end{remark}

\begin{definition}[Subobject of $\ob{C}$]
An object $\ob{S}$ and a monomorphism $\mo{m} : \ob{S} \mono \ob{C}$.
\end{definition}

\begin{remark}
Often the monomorphism can be inferred from context, so we will say $\N_+$~is a subobject of~$\Z_+$, even though technically we should say $\langle \N_+, \langle \lambda n.\; n\mathrm{~as~}\Z, \noprf, \noprf \rangle \rangle$ is a subobject of~$\Z_+$, since after all $\langle \N_+, \langle \lambda n.\; {-(n\mathrm{~as~}\Z)}, \noprf, \noprf \rangle \rangle$ is also a subobject of~$\Z_+$.
\end{remark}

\begin{definition}[$\cat{Sub}(\ob{C})$]
The category whose
\begin{framed}
\begin{description}
\item[Objects] are the subobjects of $\ob{C}$
\item[Morphisms $\langle \ob{S}_1, \mo{m}_1 \rangle \mto \langle \ob{S}_2, \mo{m}_2 \rangle$] are morphisms $\mo{m} : \ob{S}_1 \mto \ob{S}_2$ such that $\mo{m}_1 = \mo{m} \cocomp \mo{m}_2$
\item[Composition] is as in $\ob{C}$
\item[Identity] is as in $\ob{C}$
\end{description}
\end{framed}
\end{definition}

\begin{definition}[Subcategory]
A subcategory of a category~$\ob{C}$ is a subobject in~$\cat{CAT}$.
\end{definition}

\begin{example}
\begin{description}
\setlength{\itemsep}{0pt}
\item[$\cat{Set} \mono \cat{Rel}$]
\item[$\Set : \cat{Rel} \mono \cat{Set}$]
\item[$\cat{n} \mono \mathbb{\omega}$]
\item[$\mathbb{\Delta} \mono \cat{Prost}$]
\item[$\cat{Mono} \mono \cat{Alg}(2,0)$]
\item[$\cat{Prost} \mono \cat{Rel}(2)$]
\end{description}
\end{example}

\begin{definition}[Wide Subcategory]
A subcategory whose inclusion functor is surjective on objects (i.e.~$F_O$ is a surjection).
\end{definition}

\begin{definition}[Full Subcategory]
A subcategory whose inclusion functor is surjective on morphisms for each object pair (i.e.~$F_M(\ob{C}_1, \ob{C}_2)$ is a surjection for all objects $\ob{C}_1$ and~$\ob{C}_2$).
\end{definition}

\begin{exercise}
Prove that $\cat{Mon}$ is isomorphic in~$\cat{CAT}$ to the full subcategory of~$\cat{Cat}$ containing only the categories with one object.
\end{exercise}

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