\subsection{Categories}\label{subsec:categories}

\begin{definition}\label{def:category}\mcite[def. 1.1.1]{Leinster2016Basic}
  A \term{category} is a \hyperref[def:quiver]{quiver} \( \cat{C} \) equipped with a \hyperref[def:partial_function]{partial operation} \( \bincirc \) on the arrows of \( \cat{C} \) and another operation \( \id \) that selects a distinguished arrow for each vertex.

  In tradition regarding \hyperref[def:concrete_category]{forgetful functors}, we denote the underlying quiver of \( \cat{C} \) by \( U(\cat{C}) \).

  \begin{thmenum}[series=def:category]
    \thmitem{def:category/objects} We call the vertices of the quiver \term{objects} and denote the set of all objects by \( \obj(\cat{C}) \). We will often write \( A \in \cat{C} \) as a shorthand for \( A \in \obj(\cat{C}) \).

    \thmitem{def:category/morphisms} We call the arrows of the quiver \term{morphisms} or sometimes \term{maps}. If \( f \) is a morphism, we call its head its \term{domain} \( \dom(f) \) and its tail its \term{codomain} \( \co\dom(f) \). We denote a morphism from \( A \) to \( B \) by \( f: A \to B \) or \( A \reloset f \to B \).

    We call the set \( \cat{C}(A, B) \) of all morphisms from \( A \) to \( B \) a \term{morphism set} or \term{\( \hom \)-set}. We use the shorthand \( \cat{C}(A) \) for \( \cat{C}(A, A) \). Another established notation is \( \op{hom}(A, B) \) instead of \( \cat{C}(A, B) \).

    Both of these notations highlight that \( \cat{C}(A, B) \), when parameterized by \( A \) and \( B \), is a \hyperref[def:functor]{functor}, as discussed in \fullref{def:hom_functor}.

    \thmitem{def:category/composition} We require the \term{composition} \( \bincirc \) of the arrows \( f \) and \( g \) to be defined only if \( \co\dom(f) = \dom(g) \). In this case, we require \( g \bincirc f \) to be a morphism from \( \dom(f) \) to \( \co\dom(g) \).

    Note how the order of \( f \) and \( g \) may seem confusing: we write the composition of \( f: A \to B \) and \( g: B \to C \) as \( g \bincirc f: A \to C \). This is set up so that it matches \hyperref[def:multi_valued_function/composition]{function composition}. The order may seem different compared to multiplication in \hyperref[def:group]{groups}, for example, however \fullref{def:monoid_delooping} shows that this is actually a generalization of multiplication.

    This order of composition is used in \cite[7]{MacLane1994}, \cite[def. 1.1.1]{Leinster2016Basic} and \cite[def. I.3.1]{Aluffi2009}.

    \thmitem{def:category/identity} We denote the \term{identity morphism} of an object \( A \) by \( \id_A \).
  \end{thmenum}

  The definition of a category additionally requires the following conditions to hold:
  \begin{thmenum}[resume=def:category]
    \thmitem[def:category/C1]{C1} For any morphism \( f: A \to B \), the identities \( \id_A \) and \( \id_B \) must satisfy
    \begin{equation}\label{eq:def:category/C1}\tag{\logic{C1}}
      f \bincirc \id_A = \id_B \bincirc f = f.
    \end{equation}

    \thmitem[def:category/C2]{C2} Composition must be associative. That is, for each triple of morphism \( f: A \to B \), \( g: B \to C \) and \( h: C \to D \), the following must hold:
    \begin{equation}\label{eq:def:category/C2}\tag{\logic{C2}}
      (h \bincirc g) \bincirc f = h \bincirc (g \bincirc f).
    \end{equation}
  \end{thmenum}
\end{definition}

\begin{example}\label{ex:def:category}
  Examples of categories include:

  \begin{itemize}
    \item The category \( \cat{Set} \) of \hyperref[def:large_and_small_sets]{small} \hyperref[def:set]{sets} and \hyperref[def:function]{functions} defined in \fullref{def:category_of_small_sets}.

    \item The category \( \cat{Cat} \) of small categories defined in \fullref{def:category_of_small_categories}.

    \item All the \hyperref[def:category_of_small_first_order_models]{categories of small first-order models} listed in \fullref{ex:def:category_of_small_first_order_models}

    \item The category \( \cat{Top} \) of small \hyperref[def:topological_space]{topological spaces} and \hyperref[def:global_continuity]{continuous functions} defined in \fullref{def:category_of_small_topological_spaces}.

    \item For every topological space, the fundamental groupoid defined in \fullref{def:fundamental_groupoid}.

    \item The category \( \cat{Quiv} \) of small \hyperref[def:quiver]{quivers} defined in \fullref{def:category_of_small_quivers}.

    \item For every quiver, the free category defined in \fullref{def:quiver_free_category}.

    \item For every \hyperref[def:preordered_set]{preordered set}, the induced category defined in \fullref{thm:order_category_isomorphism}.
  \end{itemize}
\end{example}

\begin{definition}\label{def:category_size}
  As can be seen from \fullref{ex:def:category}, some of the categories we are working with, like \( \cat{Set} \), contain as objects all \hyperref[def:large_and_small_sets]{small sets}. As mentioned in \fullref{def:large_and_small_sets}, the concept of a small set is defined relative to the smallest Grothendieck universe that suits our needs.

  \Fullref{thm:russels_paradox} demonstrates that the set of all sets easily leads to a paradox, which is the reason we restrict our attention only to sets within some Grothendieck universe. This universe is implicit by default, however we will occasionally need to make it explicit.

  We will say that the category \( \cat{C} \) is \term{locally \( \mscrU \)-small} if the morphism set \( \cat{C}(A, B) \) is \( \mscrU \)-small for every pair of objects \( A \) and \( B \). If, in addition, the set \( \obj(\cat{C}) \) of objects is also \( \mscrU \)-small, we will say that the category \( \cat{C} \) is \term{\( \mscrU \)-small}. If a category is not \( \mscrU \)-small, we say that it is \term{\( \mscrU \)-large}.

  In particular, \term{finite} and \term{locally finite} categories are ones who are \( V_\omega \)-small and \( V_\omega \)-locally small for the universe of hereditary finite sets \hyperref[def:universe_of_hereditary_finite_sets]{\( V_\omega \)}. This notion of local finiteness is unrelated to local finiteness of graphs defined in \fullref{def:hypergraph/degree}.

  Universes are crucial to be able to do a lot of categorical constructions within set theory, most importantly \( \mscrU \)-large \hyperref[def:functor_category]{functor categories} but also \hyperref[def:product_category]{product categories} and, as discussed in \fullref{rem:functor_size}, even the \hyperref[def:functor]{functors} themselves.

  Note that, even if a category is \( \mscrU \)-small, the category itself as the tuple \( (Q, \bincirc, \id) \) from \fullref{def:category} may not be a \( \mscrU \)-small set.

  Also note that, in a locally small category, it is possible for the set of all morphisms to be \( \mscrU \)-large. This is impossible for small categories due to \ref{def:grothendieck_universe/union}.

  We sometimes skip the prefix \enquote{\( \mscrU \)-} if it is unimportant, and simply speak of \enquote{large categories} or \enquote{locally small categories}.
\end{definition}

\begin{definition}\label{def:category_of_small_sets}
  Suppose that we are given a \hyperref[def:grothendieck_universe]{Grothendieck universe} \( \mscrU \), which is safe to assume to be the smallest suitable one as explained in \fullref{def:large_and_small_sets}.

  We denote the \hyperref[def:category]{category} of \( \mscrU \)-small \hyperref[def:set]{sets} by \( \ucat{Set} \) or, if the universe is clear from the context, simply by \( \cat{Set} \). See \fullref{def:category_size} for a further discussion of universes and categories.

  \begin{itemize}
    \item The \hyperref[def:category/objects]{set of objects} \( \obj(\cat{Set}) \) is the set of all \( \mscrU \)-small sets, i.e. all members of \( \mscrU \).

    \item The \hyperref[def:category/morphisms]{set of morphisms} \( \cat{Set}(A, B) \) from \( A \) to \( B \) is the set \hyperref[def:function/set_of_functions]{\( \fun(A, B) \)} of all total single-valued functions from \( A \) to \( B \).

    \item The \hyperref[def:category/composition]{composition of morphisms} is the usual \hyperref[def:multi_valued_function/composition]{function composition}.

    \item The \hyperref[def:category/identity]{identity morphism} on the set \( A \) is the \hyperref[def:multi_valued_function/identity]{identity function}
    \begin{equation*}
      \begin{aligned}
        &\id_A: A \to A \\
        &\id_A(x) \coloneqq A.
      \end{aligned}
    \end{equation*}
  \end{itemize}
\end{definition}
\begin{defproof}
  To see that \( \ucat{Set} \) is indeed a category, we verify the conditions \ref{def:category/C1} and \ref{def:category/C2}.

  \SubProofOf{def:category/C1} For every two sets \( A, B \in \mscrU \) and every function \( f: A \to B \), for all \( x \in A \) we have
  \begin{equation*}
    [\id_B \bincirc f](x)
    =
    \id_B(f(x))
    =
    f(x)
    =
    f(\id_A(x))
    =
    [f \bincirc \id_A](x).
  \end{equation*}

  Therefore, \( \id_A \) and \( \id_B \) satisfy \eqref{eq:def:category/C1}.

  \SubProofOf{def:category/C2} Associativity of function composition is proved in \fullref{thm:def:multivalued_function/associative}.
\end{defproof}

\begin{proposition}\label{thm:category_of_small_sets_properites}
  We collect here important properties of the category \hyperref[def:category_of_small_sets]{\( \ucat{Set} \)} of \( \mscrU \)-small sets. Most of them require forward references.

  \begin{thmenum}
    \thmitem{thm:category_of_small_sets_properites/large} It is a \( \mscrU \)-large category in the sense of \fullref{def:category_size} because \( \mscrU \) itself is the set of objects and, defined as a \hyperref[def:quiver]{quiver} with additional operations, the category is a \( \mscrU \)-large set in the sense of \fullref{def:large_and_small_sets}.

    \thmitem{thm:category_of_small_sets_properites/locally_small} It is a \hyperref[def:category_size]{\( \mscrU \)-locally small category} because \( \mscrU \) is a model of \hyperref[def:zfc]{\( \logic{ZFC} \)} and \fullref{thm:zfc_existence_theorems/set_of_functions} holds.

    \thmitem{thm:category_of_small_sets_properites/morphism_invertibility} All \hyperref[def:morphism_invertibility/right_cancellative]{epimorphisms} and \hyperref[def:multi_valued_function/empty]{nonempty} \hyperref[def:morphism_invertibility/left_cancellative]{monomorphisms} \hyperref[def:morphism_invertibility/left_invertible]{split} and are precisely the \hyperref[def:function_invertibility/surjective]{surjective} and nonempty \hyperref[def:function_invertibility/injective]{injective functions}, respectively.

    This is stated in \fullref{thm:function_invertibility_categorical}. See also \fullref{thm:epimorphisms_split_in_set}.

    \thmitem{thm:category_of_small_sets_properites/universal_objects} The empty set \( \varnothing \) is an \hyperref[def:universal_objects/initial]{initial object} and the singleton set \( \set{ A } \) is a \hyperref[def:universal_objects/terminal]{terminal object} for every \( A \in \ucat{Set} \). No \hyperref[def:universal_objects/zero]{zero objects} exist in \( \ucat{Set} \) by \fullref{thm:def:universal_objects/no_zero}.

    This is discussed in \fullref{ex:def:universal_objects}.

    \thmitem{thm:category_of_small_sets_properites/discrete_category} The \hyperref[def:discrete_category]{discrete category} functor \( D: \ucat{Set} \to \ucat{Cat} \) is left adjoint to the forgetful functor \( U: \ucat{Cat} \to \ucat{Set} \)

    This is discussed in \fullref{ex:def:category_adjunction/set_cat}.

    \thmitem{thm:category_of_small_sets_properites/limits} The \hyperref[def:discrete_category_limits]{products} and \hyperref[def:discrete_category_limits]{coproducts} are the \hyperref[def:cartesian_product/product]{Cartesian products} and the \hyperref[def:disjoint_union]{disjoint unions}, respectively.

    This is stated in \fullref{thm:discrete_category_limits_in_set}.
  \end{thmenum}
\end{proposition}

\begin{definition}\label{def:opposite_category}\mcite[def. 1.1.9]{Leinster2016Basic}
  The \term{opposite} category of \( \cat{C} \) is obtained by \enquote{reversing} all arrows. This reversing is merely a relabeling of the domain and codomain --- the underlying morphisms are the same. This concept is quite powerful because it allows performing constructions and proofs by duality --- see \fullref{thm:categorical_principle_of_duality}.

  Formally, the category \( \cat{C}^{\opcat} \) is defined as follows:
  \begin{itemize}
    \item The \hyperref[def:category/objects]{set of objects} \( \obj(\cat{C}^{\opcat}) \) is the set of objects \( \obj(\cat{C}) \) of \( \cat{C}^{\opcat} \).

    \item The \hyperref[def:category/morphisms]{set of morphisms} \( \cat{C}(A, B) \) is the set \( \cat{C}(B, A) \). Thus, any morphism \( f^{\opcat}: A \to B \) in the opposite category \( \cat{C}^{\opcat} \) is a morphism \( f: B \to A \) in \( \cat{C}^{\opcat} \).

    The superscript here is used solely to distinguish between \( f \) being regarded as a morphism of \( \cat{C} \) and of \( \cat{C}^{\opcat} \) --- the morphisms in \( \cat{C} \) are exactly those of \( \cat{C}^{\opcat} \), simply relabeled.

    \item The \hyperref[def:category/composition]{composition of the morphisms}
    \begin{align*}
      f^{\opcat} &\in \cat{C}^{\opcat}(A, B) = \cat{C}(B, A) \\
      g^{\opcat} &\in \cat{C}^{\opcat}(B, C) = \cat{C}(C, B)
    \end{align*}
    is the morphism
    \begin{equation*}
      \underbrace{g^{\opcat} \bincirc f^{\opcat}}_{\cat{C}^{\opcat}(A, C)} \coloneqq \underbrace{f \bincirc g}_{\cat{C}(C, A)}.
    \end{equation*}

    \item The \hyperref[def:category/identity]{identity morphism} on the object \( A \in \cat{C} \) is again \( \id_A \).
  \end{itemize}
\end{definition}

\begin{remark}\label{rem:double_opposite_category}
  The double-opposite of a category or morphism is obviously the original. This is made precise with the oppositization functor defined in \fullref{def:opposite_functor}.
\end{remark}

\begin{example}\label{ex:def:opposite_category}
  A morphism \( f^{\opcat}: A \to B \) in the category \( \cat{Set}^{\opcat} \) is a function from the set \( B \) to the set \( A \). We cannot apply \( f \) to a point in \( B \) unless \( B \subseteq A \). Thus, we cannot regard, in general, the morphism \( f^{\opcat} \) as a function, although only the \hyperref[def:multi_valued_function]{signature} of \( f \) is different from that of \( f^{\opcat} \) --- their \hyperref[def:multi_valued_function/graph]{graphs} are the same.
\end{example}

\begin{proposition}\label{thm:categorical_principle_of_duality}
  We can extend the principle of duality for preordered sets discussed in \fullref{def:preordered_set/duality} to categories. Since we have defined categories in \hyperref[def:axiom_of_universes]{\logic{ZFC+U}} rather than as a first-order theory, we will state this principle informally:
  \begin{displayquote}
    If a statement holds for every category, its dual statement obtained by \enquote{reversing} all morphisms as in \fullref{def:opposite_category}, also holds for every category.
  \end{displayquote}

  See \fullref{thm:def:morphism_invertibility/split_epimorphism} for how this principle can be utilized easily.

  We list here results that heavily utilize this principle. Note that it is now always obvious what exactly needs to reversed in order for this principle to hold. For example, as discussed in \fullref{def:opposite_functor}, for opposite functors we have
  \begin{equation*}
    [F \bincirc G]^{\opcat} = F^{\opcat} \bincirc G^{\opcat},
  \end{equation*}
  which is somewhat unexpected.

  \begin{thmenum}
    \thmitem{thm:categorical_principle_of_duality/morphism_invertibility} \Fullref{thm:morphism_invertibility_duality}: A morphism \( f: A \to B \) in \( \cat{C} \) is a \hyperref[def:morphism_invertibility/left_invertible]{(split)} \hyperref[def:morphism_invertibility/left_cancellative]{monomorphism} if and only if \( f^{\opcat}: B \to A \) in the opposite category \( \cat{C}^{\opcat} \) is a \hyperref[def:morphism_invertibility/right_invertible]{(split)} \hyperref[def:morphism_invertibility/right_cancellative]{epimorphism}.

    In particular, \( f \) is an \hyperref[def:morphism_invertibility/isomorphism]{isomorphism} in \( \cat{C} \) if and only if \( f^{\opcat} \) is an isomorphism in \( \cat{C}^{\opcat} \).

    \thmitem{thm:categorical_principle_of_duality/universal_objects} \Fullref{thm:universal_object_duality}: An object is \hyperref[def:universal_objects/initial]{initial} if and only if it is a \hyperref[def:universal_objects/terminal]{terminal object} the \hyperref[def:opposite_category]{opposite category}.

    \thmitem{thm:categorical_principle_of_duality/functor_categories} \Fullref{thm:opposite_of_functor_category}: For the \hyperref[def:opposite_category]{opposite} of the \hyperref[def:functor_category]{functor category} \( [\cat{C}, \cat{D}] \) we have
    \begin{equation*}
      [\cat{C}, \cat{D}]^{\opcat} = [\cat{C}^{\opcat}, \cat{D}^{\opcat}].
    \end{equation*}

    \thmitem{thm:categorical_principle_of_duality/equivalences} \Fullref{thm:opposite_of_category_equivalence}: The \hyperref[def:opposite_category]{duals} of \hyperref[def:category_equivalence]{equivalent categories} are equivalent.

    \thmitem{thm:categorical_principle_of_duality/adjunctions} \Fullref{thm:category_adjunction_duality}: The functor \( F \) is \hyperref[def:category_adjunction]{left adjoint} to \( G \) if and only if the \hyperref[def:opposite_functor]{dual functor} \( F^{\opcat} \) is right adjoint to \( G^{\opcat} \).

    \thmitem{thm:categorical_principle_of_duality/limits} \Fullref{thm:categorical_limit_duality}: For every \hyperref[def:category_of_cones/cone]{cone} \( (A, \alpha) \) of the \hyperref[def:categorical_diagram]{diagram} \( D \) in \( \cat{C} \), \( (A, \alpha^{\opcat}) \) is a \hyperref[def:category_of_cones/cone]{cocone} of \( D^{\opcat} \) in \( \cat{C}^{\opcat} \).

    Even more, for every \hyperref[def:category_of_cones/limit]{limit} \( (L, \pi) \) of \( D \) in \( \cat{C} \), \( (L, \pi^{\opcat}) \) is a \hyperref[def:category_of_cones/colimit]{colimit} of \( D^{\opcat} \) in \( \cat{C}^{\opcat} \).
  \end{thmenum}
\end{proposition}

\begin{definition}\label{def:morphism_invertibility}
  In connection with \fullref{def:function_invertibility} and \fullref{def:first_order_homomorphism_invertibility}, we introduce the following terminology:
  \begin{thmenum}
    \thmitem{def:morphism_invertibility/left_cancellative} The morphism \( g: B \to C \) is \term{left-cancellative} if, for any pair of morphisms \( f_1, f_2: A \to B \), the equality \( g \bincirc f_1 = g \bincirc f_2 \) implies \( f_1 = f_2 \).

    Left-cancellative morphisms are also called \term{monic morphisms} or \term{monomorphisms}.

    \thmitem{def:morphism_invertibility/left_invertible} The morphism \( f: A \to B \) is \term{left-invertible} if there exists a morphism \( g: B \to A \) such that \( g \bincirc f = \id_A \). We call \( g \) a \term{left inverse} of \( f \).

    Using forward references to \fullref{def:categorical_diagram}, we can restate this condition by saying that the following diagram commutes:
    \begin{equation}\label{eq:def:morphism_invertibility/left_invertible}
      \begin{aligned}
        \includegraphics[page=1]{output/def__morphism_invertibility.pdf}
      \end{aligned}
    \end{equation}

    Left-invertible morphisms are sometimes called \term{split monomorphisms} because they \enquote{split} the identity \( \id_A \) into a composition of \( f \) and \( g \).

    \thmitem{def:morphism_invertibility/right_cancellative} \hyperref[thm:categorical_principle_of_duality]{Dually}, the morphism \( f: A \to B \) is \term{right-cancellative} if, for any pair of morphisms \( g_1, g_2: B \to C \), the equality \( g_1 \bincirc f = g_2 \bincirc f \) implies \( g_1 = g_2 \).

    Right-cancellative morphisms are also called \term{epic morphisms} or \term{epimorphisms}.

    \thmitem{def:morphism_invertibility/right_invertible} The morphism \( g: B \to A \) is \term{right-invertible} if there exists a morphism \( f: A \to B \) such that \( f \bincirc g = \id_B \). We call \( g \) a \term{right inverse} of \( f \).

    Using forward references to \fullref{def:categorical_diagram}, we can restate this condition by saying that the following diagram commutes:
    \begin{equation}\label{eq:def:morphism_invertibility/right_invertible}
      \begin{aligned}
        \includegraphics[page=2]{output/def__morphism_invertibility.pdf}
      \end{aligned}
    \end{equation}

    Right-invertible morphisms are sometimes called \term{split epimorphisms} because they \enquote{split} the identity \( \id_B \) into a composition of \( g \) and \( f \).

    \thmitem{def:morphism_invertibility/isomorphism} The morphism \( f: A \to B \) is \term{fully invertible} it is both left-invertible and right-invertible. By \fullref{thm:def:morphism_invertibility/left_and_right}, in this case, there exists a unique morphism \( f^{-1}: B \to A \) that is a \term{two-sided inverse}, i.e. it is both a left inverse and a right inverse.

    A fully invertible morphism is usually called an \term{isomorphism}. If there exists an isomorphism between \( A \) and \( B \), we say that they are \term{isomorphic} and write \( A \cong B \).

    \thmitem{def:morphism_invertibility/endomorphism} A morphism \( f: A \to A \) from an object to itself is called an \term{endomorphism}.

    \thmitem{def:morphism_invertibility/automorphism} A morphism that is both an endomorphism and an isomorphism is called an \term{automorphism}.
  \end{thmenum}
\end{definition}

\begin{example}\label{ex:def:morphism_invertibility}
  \Fullref{thm:function_invertibility_categorical} characterizes the cancellative and invertible morphisms defined in \fullref{def:morphism_invertibility} for \hyperref[def:category_of_small_sets]{\( \cat{Set} \)} in terms of \hyperref[def:function_invertibility/injective]{injectivity} and \hyperref[def:function_invertibility/injective]{surjectivity}.

  A very simple example of a monomorphism which does not split is the empty function with nonempty domain. These are discussed in \fullref{thm:function_invertibility_categorical/empty}.

  \Fullref{thm:surjective_functions_are_right_invertible} is important enough to have a categorical interpretation via \fullref{thm:epimorphisms_split_in_set}, where its relation to the \hyperref[def:zfc/choice]{axiom of choice} is also discussed.
\end{example}

\begin{proposition}\label{thm:morphism_invertibility_duality}
  A morphism \( f: A \to B \) in \( \cat{C} \) is a \hyperref[def:morphism_invertibility/left_invertible]{(split)} \hyperref[def:morphism_invertibility/left_cancellative]{monomorphism} if and only if \( f^{\opcat}: B \to A \) in the opposite category \( \cat{C}^{\opcat} \) is a \hyperref[def:morphism_invertibility/right_invertible]{(split)} \hyperref[def:morphism_invertibility/right_cancellative]{epimorphism}.

  In particular, \( f \) is an \hyperref[def:morphism_invertibility/isomorphism]{isomorphism} in \( \cat{C} \) if and only if \( f^{\opcat} \) is an isomorphism in \( \cat{C}^{\opcat} \).

  This is part of the duality principles listed in \fullref{thm:categorical_principle_of_duality}.
\end{proposition}
\begin{proof}
  Trivial.
\end{proof}

\begin{proposition}\label{thm:def:morphism_invertibility}
  Morphisms have the following basic properties regarding their \hyperref[def:morphism_invertibility]{invertibility} (compare to \fullref{thm:function_composition_invertibility}):

  \begin{thmenum}
    \thmitem{thm:def:morphism_invertibility/split_monomorphism} Any \hyperref[def:morphism_invertibility/left_invertible]{left-invertible morphism} is \hyperref[def:morphism_invertibility/left_cancellative]{left-cancellative}.

    In more categorical terms, every split monomorphism is a monomorphism.

    \thmitem{thm:def:morphism_invertibility/split_epimorphism} Any \hyperref[def:morphism_invertibility/right_invertible]{right-invertible morphism} is \hyperref[def:morphism_invertibility/right_cancellative]{right-cancellative}.

    In more categorical terms, every split epimorphism is an epimorphism.

    \thmitem{thm:def:morphism_invertibility/at_most_one_inverse}\mcite[exer. 1.1.13]{Leinster2016Basic} Any morphism has at most one two-sided inverse.

    \thmitem{thm:def:morphism_invertibility/left_and_right} If a morphism is both left-invertible and right-invertible, the two inverses are equal, and the morphism is fully invertible.

    \thmitem{thm:def:morphism_invertibility/inverse_interchanges} The morphism \( f: A \to B \) is a right inverse of \( g: B \to A \) if and only if \( g \) is a left inverse of \( f \).

    \thmitem{thm:def:morphism_invertibility/monomorphism_and_split_epimorphism} If a morphism left-cancellative and right-invertible, it is an isomorphism.

    \thmitem{thm:def:morphism_invertibility/split_monomorphism_and_epimorphism} If a morphism left-invertible and right-cancellative, it is an isomorphism.

    \thmitem{thm:def:morphism_invertibility/cancellative_composition} The composition of two monomorphisms (resp. epimorphisms) is again a monomorphism (resp. epimorphism).

    \thmitem{thm:def:morphism_invertibility/invertible_composition} The composition of two split monomorphisms (resp. epimorphisms) is again a split monomorphism (resp. epimorphism).
  \end{thmenum}
\end{proposition}
\begin{proof}
  \SubProofOf{thm:def:morphism_invertibility/split_monomorphism} Suppose that \( g: B \to C \) is left-invertible with inverse \( h: C \to B \). Suppose that \( f_1, f_2: A \to B \) are morphisms such that
  \begin{equation*}
    g \bincirc f_1 = g \bincirc f_2.
  \end{equation*}

  Then
  \begin{equation*}
    f_1
    \reloset {\eqref{eq:def:category/C1}} =
    \id_B \bincirc f_1
    =
    (h \bincirc g) \bincirc f_1
    \reloset {\eqref{eq:def:category/C2}} =
    h \bincirc (g \bincirc f_1)
    =
    h \bincirc (g \bincirc f_2)
    =
    \cdots
    =
    f_2.
  \end{equation*}

  \SubProofOf{thm:def:morphism_invertibility/split_epimorphism} This is an exemplar proof using duality. By \fullref{thm:morphism_invertibility_duality}, every split epimorphism \( f: A \to B \) in \( \cat{C} \) is a split monomorphism in \( \cat{C}^{\opcat} \). By \fullref{thm:def:morphism_invertibility/split_monomorphism}, \( f^{\opcat} \) is a monomorphism. Then gain by \fullref{thm:morphism_invertibility_duality}, \( f \) is an epimorphism.

  \SubProofOf{thm:def:morphism_invertibility/at_most_one_inverse} If \( f: A \to B \) has no inverse, it vacuously has at most one inverse.

  Now assume that \( f: A \to B \) has two inverses \( g_1: B \to A \) and \( g_2: B \to A \):
  \begin{align*}
    g_1 \bincirc f = \id_A &&& f \bincirc g_1 = \id_B, \\
    g_2 \bincirc f = \id_A &&& f \bincirc g_2 = \id_B.
  \end{align*}

  Then
  \begin{equation*}
    g_1
    \reloset {\eqref{eq:def:category/C1}} =
    g_1 \bincirc \id_B
    =
    g_1 \bincirc (f \bincirc g_2)
    \reloset {\eqref{eq:def:category/C2}} =
    (g_1 \bincirc f) \bincirc g_2
    =
    \id_A \bincirc g_2
    \reloset {\eqref{eq:def:category/C1}} =
    g_2.
  \end{equation*}

  \SubProofOf{thm:def:morphism_invertibility/left_and_right} Suppose that \( f: A \to B \) has a left-inverse \( l: B \to A \) and a right-inverse \( r: B \to A \). Then

  \SubProofOf{thm:def:morphism_invertibility/inverse_interchanges} Trivial.

  \SubProofOf{thm:def:morphism_invertibility/monomorphism_and_split_epimorphism} Let \( g: B \to A \) be left-cancellative and right-invertible. Let \( f: A \to B \) be a right inverse of \( g \). Then
  \begin{equation*}
    f
    =
    \reloset {\eqref{eq:def:category/C1}} =
    f \bincirc \id_A
    =
    f \bincirc (g \bincirc f)
    =
    (f \bincirc g) \bincirc f.
  \end{equation*}

  Because \( g \) is a left inverse of \( f \), from \fullref{thm:def:morphism_invertibility/split_monomorphism} it follows that \( f \) is left-cancellative. Since we have
  \begin{equation*}
    \id_B \bincirc f
    =
    (f \bincirc g) \bincirc f,
  \end{equation*}
  it follows that \( f \bincirc g = \id_B \).

  Therefore, \( f \) is a left inverse of \( g \) and hence an isomorphism.

  \SubProofOf{thm:def:morphism_invertibility/split_monomorphism_and_epimorphism} The proof is analogous to \fullref{thm:def:morphism_invertibility/monomorphism_and_split_epimorphism}.

  \SubProofOf{thm:def:morphism_invertibility/cancellative_composition} Let \( g: B \to C \) and \( h: C \to D \) be monomorphisms (left-cancellative).

  Let \( f_1, f_2: A \to B \) be two arbitrary morphisms with codomain \( B \). Suppose that
  \begin{equation*}
    (h \bincirc g) \bincirc f_1 = (h \bincirc g) \bincirc f_2.
  \end{equation*}

  Then, by \ref{def:category/C2},
  \begin{equation*}
    h \bincirc (g \bincirc f_1) = h \bincirc (g \bincirc f_2).
  \end{equation*}

  Since \( h \) is left-cancellative, it follows that
  \begin{equation*}
    g \bincirc f_1 = g \bincirc f_2.
  \end{equation*}

  Since \( g \) is also left-cancellative, \( f_1 = f_2 \).

  Therefore, \( h \bincirc g \) is a monomorphism.

  The proof for composition of epimorphisms is identical.

  \SubProofOf{thm:def:morphism_invertibility/invertible_composition} Let \( f: A \to B \) and \( g: B \to C \) be split monomorphisms (left-invertible).

  Then there exist left inverses \( l_f: B \to A \) and \( l_g: C \to B \) of \( f \) and \( g \), respectively. We have
  \begin{equation*}
    (l_f \bincirc l_g) \bincirc (g \bincirc f)
    \reloset {\eqref{eq:def:category/C2}} =
    l_f \bincirc (l_g \bincirc g) \bincirc f
    =
    l_f \bincirc \id_B \bincirc f
    \reloset {\eqref{eq:def:category/C1}} =
    l_f \bincirc f
    =
    \id_A.
  \end{equation*}

  Therefore, \( g \bincirc f \) is also left-invertible.

  The proof for composition of split epimorphisms is identical.
\end{proof}

\begin{theorem}[Epimorphisms split in Set]\label{thm:epimorphisms_split_in_set}
  Every \hyperref[def:morphism_invertibility/right_cancellative]{epimorphism} in \hyperref[def:category_of_small_sets]{\( \cat{Set} \)} splits. That is, all epimorphisms in \( \cat{Set} \) are \hyperref[def:morphism_invertibility/right_invertible]{split epimorphisms}.

  Assuming the existence of the \hyperref[def:grothendieck_universe]{Grothendieck universe} containing \( \cat{Set} \), in \hyperref[def:zfc]{\logic{ZF}} this theorem is equivalent to the \hyperref[def:zfc/choice]{axiom of choice} --- see \fullref{thm:axiom_of_choice_equivalences/epimorphisms}.

  Since not every epimorphism splits in a general category, this theorem is sometimes considered to be a categorical statement of the axiom of choice, which holds in some categories but not in others.
\end{theorem}
\begin{proof}
  By \fullref{thm:function_invertibility_categorical/right_cancellative}, a function is an epimorphism if and only if it is surjective. Thus, the theorem is equivalent to \fullref{thm:surjective_functions_are_right_invertible}.
\end{proof}

\begin{definition}\label{def:universal_objects}\mcite[def. 2.1.7]{Leinster2016Basic}
  Fix a category \( \cat{C} \).

  \begin{thmenum}
    \thmitem{def:universal_objects/initial} We call the object \( I \in \cat{C} \) an \term{initial object} if for any other object \( A \in \cat{C} \) there exists a unique morphism \( f: I \to A \),

    \thmitem{def:universal_objects/terminal} \hyperref[thm:categorical_principle_of_duality]{Dually}, we call the object \( T \in \cat{C} \) a \term{terminal object} or \term{final object} if for any other object \( A \in \cat{C} \) there exists a unique morphism \( f: A \to T \).

    The initial and terminal objects are collectively called \term{universal objects}.

    \thmitem{def:universal_objects/zero}\mcite{nLab:pointed_category} If \( Z \) is both an initial and a terminal object, we say that \( Z \) is a \term{zero object}. A category with a zero object is called a \term{pointed category}.
  \end{thmenum}
\end{definition}

\begin{example}\label{ex:def:universal_objects}
  \begin{thmenum}
    \thmitem{ex:def:universal_objects/set} In the category \hyperref[def:category_of_small_sets]{\( \cat{Set} \)} of small sets, for any set \( A \) there is a unique \hyperref[def:multi_valued_function/empty]{empty function} from \( \varnothing \) to \( A \). Therefore, \( \varnothing \) is an \hyperref[def:universal_objects/initial]{initial object} in \( \cat{Set} \).

    For any set \( A \), there is a unique function that contracts any set \( B \) to \( \set{ A } \). Therefore, every singleton set is a \hyperref[def:universal_objects/terminal]{final object} in \( \cat{Set} \).

    We often denote the initial and terminal objects in \( \cat{Set} \) by \( 0 \) and \( 1 \) respectively, which corresponds to their definition as \hyperref[def:ordinal]{ordinals}.

    By \fullref{thm:def:universal_objects/no_zero}, \( \cat{Set} \) has no zero object.

    \thmitem{ex:def:universal_objects/grp} In the category \hyperref[def:group/category]{\( \cat{Grp} \)} of small groups, the \hyperref[def:group/trivial]{trivial group} is a \hyperref[def:universal_objects/zero]{zero object}. This holds more generally for pointed sets rather than groups.

    Indeed, it can be embedded into any other group and any group can be contracted into the corresponding trivial group. Furthermore, all trivial groups are isomorphic.
  \end{thmenum}
\end{example}

\begin{proposition}\label{thm:universal_object_duality}
  An object is \hyperref[def:universal_objects/initial]{initial} if and only if it is a \hyperref[def:universal_objects/terminal]{terminal object} the opposite category.

  This is part of the duality principles listed in \fullref{thm:categorical_principle_of_duality}.
\end{proposition}
\begin{proof}
  Trivial.
\end{proof}

\begin{proposition}\label{thm:def:universal_objects}
  \hfill
  \begin{thmenum}
    \thmitem{thm:def:universal_objects/initial} An \hyperref[def:universal_objects/initial]{initial object} is unique up to an isomorphism.
    \thmitem{thm:def:universal_objects/terminal} \hyperref[thm:categorical_principle_of_duality]{Dually}, a \hyperref[def:universal_objects/initial]{terminal object} is also unique up to an isomorphism.
    \thmitem{thm:def:universal_objects/zero} If a category has an initial and a terminal object and if they are isomorphic, then both are zero objects.

    In particular, a zero object is unique up to an isomorphism.

    \thmitem{thm:def:universal_objects/no_zero} If an initial and a terminal object exists and are not isomorphic, then there exist no zero objects.
  \end{thmenum}
\end{proposition}
\begin{proof}
  \SubProofOf{thm:def:universal_objects/initial} Suppose that \( A \) and \( B \) are both initial objects in \( \cat{C} \). Then there exist morphisms \( f: A \to B \) and \( g: B \to A \). Their composition \( g \bincirc f \) is an \hyperref[def:morphism_invertibility/endomorphism]{endomorphism} on \( A \).

  But there exists a unique \hyperref[def:morphism_invertibility/endomorphism]{endomorphism} on \( A \), which must be the identity \( \id_A \). Thus, \( g \bincirc f = \id_A \) and \( g \) is a left inverse of \( f \).

  We can analogously show that \( g \) is a right inverse of \( f \). Therefore, \( f \) is fully invertible, and \( A \) and \( B \) are isomorphic.

  \SubProofOf{thm:def:universal_objects/terminal} If \( T' \) and \( T^\dprime \) are terminal objects in \( \cat{C} \), by \fullref{thm:universal_object_duality}, they are initial objects in \( \cat{C}^{\opcat} \). By \fullref{thm:def:universal_objects/initial}, they are isomorphic in \( \cat{C}^{\opcat} \) and by \fullref{thm:morphism_invertibility_duality}, they are isomorphic in \( \cat{C} \).

  \SubProofOf{thm:def:universal_objects/zero} Suppose that \( A \) is an initial object and that \( B \) is a final object in \( \cat{C} \). Let \( f: A \to B \) be an isomorphism between them.

  Let \( C \in \cat{C} \) be any other object and let \( g: C \to B \) be the unique morphism to \( B \). Then \( f^{-1} \bincirc g: C \to A \) is a morphism from \( C \) to \( A \). The inverse \( f^{-1}: B \to A \) is unique by \fullref{thm:def:morphism_invertibility/at_most_one_inverse}, therefore its composition with \( g: C \to B \) is also unique. Hence, any object has a unique morphism to \( A \). This makes \( A \) a terminal object and thus a zero object.

  We can analogously show that \( B \) is a zero object.

  \SubProofOf{thm:def:universal_objects/no_zero} By \fullref{thm:def:universal_objects/zero}, all zero objects are isomorphic. By \fullref{thm:def:universal_objects/initial}, all initial objects are isomorphic and analogously for terminal objects. Hence, if a zero object exists, all initial objects are isomorphic to all terminal objects.

  If some initial object is not isomorphic to some terminal object, then by contraposition it follows that no zero object exists.
\end{proof}


\section{Orbits}
The study of orbits it is motivated by the motion of heavenly bodies under the infludence of the gravitational force due to e.g. a star.
Of course, we want to study a conservative field $-\nabla V$ where $V$ is a potential of a central force (so it only depends on $r=|\underline{r}|$), that is
$$m\underline{\ddot{r}}=-\nabla V(r)$$
We shall study the case when the central body is much more massive than the orbiting body, so that the central body can be regarded as fixed.\\
Recall that $\underline{L}=m\underline{r}\times\underline{\dot{r}}$, and in the case for a central force $\underline{\dot{L}}=0$, so $\underline{L}$ is constant.
Also we always have $\underline{L}\cdot\underline{r}=0$, so we can regard the motion as if it is in a plane.
\subsection{Polar Coordinates in a Plane}
In an orbit problem, we of course want to use polar coordinate to simplify calculation.
Since we can regard the problem as two-dimensional, we can use the plane polar coordinates $x=r\cos\theta,y=r\sin\theta$, so we define the unit vectors
$$\underline{e_r}=(\cos\theta,\sin\theta)^\top,\underline{e_\theta}=(-\sin\theta,\cos\theta)^\top$$
So we can use $\underline{e_r},\underline{e_\theta}$ as two basis vectors, but note that they are dependent of the position.
Note that $\underline{e_r}$ is always in the direction of the position and $\underline{e_\theta}$ the direction of rotation.
Also note that $\mathrm d\underline{e_r}/\mathrm d\theta=\underline{e_\theta},\mathrm d\underline{e_\theta}/\mathrm d\theta=-\underline{e_r}$, so
$$\frac{\mathrm d\underline{e_r}}{\mathrm dt}=\underline{e_\theta}\dot{\theta},\frac{\mathrm d\underline{e_\theta}}{\mathrm dt}=-\underline{e_r}\dot{\theta}$$
Now we turn to consider the implications for the velocity of the particle given some acceleration.
Write $\underline{r}=r\underline{e_r}$.
Consider this as a function of time, then $\underline{v}=\underline{\dot{r}}=\dot{r}\underline{e_r}+r\underline{e_\theta}\dot{\theta}$.
$\dot{r}$ is the radial component of the velocity while $\dot{\theta}$ is the angular component of it.
So $\dot{\theta}$ has dimension $T^{-1}$.\\
As for accelerations, we have
$$\underline{\ddot{r}}=\underline{\dot{v}}=\ddot{r}\underline{e_r}+\dot{r}\underline{e_\theta}\dot{\theta}+(\dot{r}\dot{\theta}+r\ddot{\theta})\underline{e_\theta}-r\dot{\theta}\underline{e_r}\dot{\theta}=(\ddot{r}-r\dot{\theta}^2)\underline{e_r}+(2\dot{r}\dot{\theta}+r\ddot{\theta})\underline{e_\theta}$$
\begin{example}
    Consider the circular motion with a constant angular velocity, then $r=a,\dot{\theta}=\omega$, so $\dot{r}=\ddot{\theta}=0$, hence
    $$\underline{\ddot{r}}=(\ddot{r}-r\dot{\theta}^2)\underline{e_r}+(2\dot{r}\dot{\theta}+r\ddot{\theta})\underline{e_\theta}=-a\omega^2\underline{e_r}=-\omega^2\underline{r}$$
    which is the familiar centripetal acceleration.
    Newton's Second Law requires a force to be applied to cause this acceleration, which is called the centripetal force, which is in the direction of $-\underline{e_r}$.
    For the special case that it is actually a mass on a string, then when the string broke, the mass will move in a straight line that is tangential to the point where the string broke.
\end{example}
\subsection{Motion in a Constant Force Field}
We know
$$m\underline{\ddot{r}}=\underline{F}=-\nabla V(r)=-\frac{\mathrm dV}{\mathrm dr}\underline{e_r}$$
for a force field that is symmetric wrt the origin.
Note that
$$-\frac{\mathrm dV}{\mathrm dr}\underline{e_r}=\underline{F}=m(\ddot{r}-r\dot{\theta}^2)\underline{e_r}+m(2\dot{r}\dot{\theta}+r\ddot{\theta})\underline{e_\theta}$$
By looking at the $\underline{e_\theta}$ component, we have $2\dot{r}\dot{\theta}+r\ddot{\theta}=0$, so
$$\frac{1}{r}\frac{\mathrm d}{\mathrm dt}(mr^2\dot{\theta})=0$$
Hence the quantity $mr^2\dot{\theta}$ is constant, but $\underline{L}=\underline{r}\times (m\underline{\dot{r}})=mr^2\dot{\theta}\underline{e_z}$.
Where $\underline{e_z}$ is the normal to the plane of motion.
Hence the angular momentum is constant in magnitude.
We write $h=|\underline{L}|/m=r^2\dot{\theta}$.\\
Going to the radial part $\mathrm dV/\mathrm dr=-m(\ddot{r}-h^2/r^3)$, rearranging gives
$$m\ddot{r}=-\frac{\mathrm dV}{\mathrm dr}+\frac{mh^2}{r^3}=-\frac{\mathrm dV_{\rm eff}}{\mathrm dr},V_{\rm eff}=V+\frac{mh^2}{2r^2}$$
So the motion of the particle is as if we are considering one dimensional motion under the influence of a modified potential $V_{\rm eff}$.\\
The energy of the particle is then
$$E=T+V=\frac{1}{2}m|\underline{\dot{r}}|^2+V(r)=\frac{1}{2}m\dot{r}^2+V_{\rm eff}(r)$$
\begin{example}
    For gravity, we have
    $$V(r)=-\frac{GMm}{r},V_{\rm eff}(r)=-\frac{GMm}{r}+\frac{mh^2}{2r^2}$$
    So $V_{\rm eff}$ is minimum at $r=h^2/GM$ and minimum energy is
    $$E_{\rm min}=-m(GM)^2/(2h^2)$$
    At the minimum (which is a stable equilibrium), both $r,\dot\theta$ are constants.\\
    At any $E_{\rm min}<E<0$, the particle oscillates.
    Let $r_0$ be the point where $V_{\rm eff}=0$, then $r_0<r_{\rm min}\le r\le r_{\rm max}$.
    So it gives a bounded non-circular orbit with $\dot\theta$ varying when $r$ varies.
    $r_{\rm min}$ is called the periapsis and $r_{\rm max}$ is called apoapsis.\\
    For $E>0$, the particle can move from a long distance and escape, which is an unbounded orbit.
\end{example}
\subsection{Stability of Circular Orbits}
Consider the potential $V(r)$, we want to investigate whether a circular orbit exists and whether it is stable.\\
Assume that the angular momentum $h$ is given and is nonzero.
For circular orbit $r(t)=r_\star$ is a constant, then
$$\ddot{r}=0\implies V_{\rm eff}^\prime(r_\star)=0$$
which is the condition for circular orbit.
Note that if $V_{\rm eff}^{\prime\prime}(r_\star)>0$, then it is a minimum, so $r_\star$ is a stable fixed point.
So if we express it as $V(r)$, we get
$$0=V_{\rm eff}^\prime(r_\star)=V^\prime(r_\star)-\frac{mh^2}{r_\star^3}=0\implies V^\prime(r_\star)=\frac{mh^2}{r_\star^3}$$
And it is stable if
$$0<V_{\rm eff}^\prime(r_\star)=V^{\prime\prime}(r_\star)+\frac{3mh^2}{r_\star^4}=V^{\prime\prime}(r_\star)+\frac{3V^\prime(r_\star)}{r_\star}$$
So in terms of $F(r)$, we have
$$F^{\prime}(r_\star)+\frac{3F(r_\star)}{r_\star}<0$$
\begin{example}
    If we take $V(r)=-km/rp$ for $k,p>0$ for a circular orbit with radius $r_\star$, we can solve the above equation to get $r_\star=(pk/h^2)^{1/(p-2)}$.
    So unless $p=2$, there exists a circular orbit.\\
    As for stability, we have
    $$V^{\prime\prime}(r_\star)+\frac{3V^\prime(r_\star)}{r_\star}=\frac{p(2-p)k}{r_\star^{p+2}}>0$$
    which is positive iff $p<2$.
\end{example}
\subsection{The Orbit Equation}
The shape of the orbit is obviously governed by the joint variation of $r$ and $\theta$ (both as functions of $t$).
In principle, the energy equation can be helpful to determine $r(t)$, i.e.
$$E=\frac{1}{2}m\dot{r}^2+V_{\rm eff}(r)\implies t=\pm\sqrt{\frac{m}{2}}\int\frac{\mathrm dr}{\sqrt{E-V_{\rm eff}(r)}}$$
Given $r(t)$, since we already know the conservation of angular momentum $r^2\dot\theta=h$, we can then deduce $\theta(t)$.
But this might not always yield an analytic solution.\\
An interesting approach if one is only interested in the trajectory is to use $\theta$ as the dependent variable.
We can write
$$\frac{\mathrm d}{\mathrm dt}=\dot\theta\frac{\mathrm d}{\mathrm d\theta}=\frac{h}{r^2}\frac{\mathrm d}{\mathrm d\theta}$$
So plugging in Newton's Second Law,
$$m\frac{h}{r^2}\frac{\mathrm d}{\mathrm d\theta}\left( \frac{h}{r^2}\frac{\mathrm dr}{\mathrm d\theta} \right)-\frac{mh^2}{r^3}=F(r)$$
which then becomes, by substituting $u=1/r$,
$$\frac{\mathrm d^2u}{\mathrm d\theta^2}+u=-\frac{1}{mh^2u^2}F\left(\frac{1}{u}\right)$$
This is called the orbit equation.
We can then solve for $u$ as a function of $\theta$, and then $\dot\theta=hu^2$ can help us to deduce the time evolution.
\subsection{The Kepler Problem}
We want to solve the case for gravitational central force given by
$$F(r)=-\frac{mk}{r^2}$$
So the orbit equation becomes
$$\frac{\mathrm d^2u}{\mathrm d\theta^2}+u=\frac{k}{h^2}$$
Which is linear in $u$.
We know how to solve this.
Indeed, the general solution is given by
$$u=\frac{k}{h^2}+A\cos(\theta-\theta_0)$$
WLOG we assume $A\ge 0$.\\
If $A=0$, then $u$ is constant hence we obtain a circular orbit.
If $A>0$, $u$ obtains its maximum (hence $r$ obtains its minimum) at $\theta=\theta_0$.
We may choose $\theta_0=0$, then
$$r=\frac{1}{u}=\frac{\ell}{1+e\cos\theta},\ell=\frac{h^2}{k},e=\frac{Ah^2}{k}$$
Which is the polar coordinate form of a conic section with focus at the origin.
$e$ is called the eccentricities, which determines the shape of the trajectory.
By rearranging we obtain (since $r=\ell-ex$ and $r\cos\theta=y$)
$$(1-e^2)x^2+2elx+y^2=\ell^2$$
Therefore if $e\in[0,1)$, it is an ellipse that is bounded by
$$\frac{\ell}{1+e}\le r\le\frac{\ell}{1-e}$$
Or analytically we can rewrite the equation as
$$\frac{(x+ea)^2}{a^2}+\frac{y^2}{b^2}=1,a=\frac{\ell}{1-e^2},b=\frac{\ell}{\sqrt{1-e^2}}\le a$$
$a,b$ represents the semimajor and semiminor axes respectively.
In particular, for $e=0$, the path is a circle with center being the central mass.\\
For $e>1$, the equation gives a hyperbola, so $r\to\infty$ when $\theta\to\pm\alpha$ where $\alpha=\cos^{-1}(-1/e)\in (\pi/2,\pi)$.
We can also transform the equation ot the standard equation for hyperbola
$$\frac{(x-ea)^2}{a^2}-\frac{y^2}{b^2}=1$$
with $a=\ell/(e^2-1),b=\ell/\sqrt{e^2-1}$.
This case represents incoming body with large velocity which is deflected by gravitational force.
By simple calculations the asymptotes are $y=\mp b(x-ea)/a$, so $bx\pm ay=eba$.
And the normal vectors are $\underline{n}=(b,\pm a)/\sqrt{a^2+b^2}$.\\
Now consider the perpendicular distance between incoming mass and orgin, we have
$$\underline{r}\cdot\underline{n}=(x,y)\cdot\left( \frac{b}{\sqrt{a^2+b^2}},\pm\frac{a}{\sqrt{a^2+b^2}} \right)=\frac{eba}{\sqrt{a^2+b^2}}=b$$
This is sometimes called the impact parameter.\\
The marginal case that $e=1$ yields a parabola with equation
$$r=\frac{\ell}{1+\cos\theta}$$
where $r\to \infty$ as $\theta\to\pm\pi$.
In Cartesians this reduces to $y=2\ell(\ell-x)$.\\
On the other hand, we might want to analyze the linkge between the energy and the eccentricity of the trajectory.
Recall that
\begin{align*}
    E&=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-\frac{mk}{r}\\
    &=\frac{1}{2}mh^2\left( \left( \frac{\mathrm du}{\mathrm d\theta} \right)^2+u^2 \right)-mku\\
    &=\frac{mk}{2\ell}(e^2-1)
\end{align*}
Hence bounded orbits have $e<1,E<0$ and unbounded ones have $e>1,E>0$.
The marginal case is then $e=E=0$.
\begin{law}[Kepler's Laws of Planetary Motion]
    1. Orbit of planet is ellipe with the Sun at focus.\\
    2. Line between the planet and the sun sweeps cut equal area in equal time.\\
    3. Square of period $P$ is proportional to cube of semimajor axis.
\end{law}
1 is consistent with the solutrion to the orbit equation that we have obtained earlier, and $2$ follows from the conservation of angular momentum (since the rate of change of area is approximately $r^2\dot\theta/2=h/2$).
Hence the area of the ellipse is $A=hP/2$ where $P$ is the period.
Therefore $\pi ab=hP/2$, rearranging gives the third statement.
\subsection{Rutherford Scattering}
Consider the motion in a repulsive force under inverse square law:
$$V(r)=\frac{mk}{r},F(r)=\frac{mk}{r^2}$$
Then the orbit equation solves to give
$$\frac{1}{r}=u=-\frac{k}{h^2}+A\cos(\theta-\theta_0)$$
WLOG $\theta_0=0,A\ge 0$.
So
$$r=\frac{\ell}{e\cos\theta-1},\ell=\frac{h^2}{k},e=\frac{Ah^2}{k}$$
If there is sometime where $r>0$, then we necessarily have $e>1$, therefore the trajectory is a hyperbola.
As previously known, $r\to\infty$ as $\theta\to\pm\alpha$ where $\alpha=\cos^{-1}(1/e)\in(0,\pi/2)$ and in Cartesian,
$$\frac{(x-ea)^2}{a^2}-\frac{y^2}{b^2}=1,a=\frac{\ell}{e^2-1},b=\frac{\ell}{\sqrt{e^2-1}}$$
Suppose the speed of the particle from far away be $v$, that is
With $x$-axis parallel to the incoming asymptote, as $t\to-\infty$
$$\underline{r}(t)\to (x(t),b,0),\underline{\dot{r}}(t)\to(-v,0,0)$$
Then $\underline{r}\times\underline{\dot{r}}\to(0,0,bv)$
Therefore the angular momentum per unit mass is $bv$, so
$$b=\frac{h^2}{k}\frac{k}{bv^2}=\frac{h^2}{k}\tan\frac{\beta}{2}=\frac{b^2v^2}{k}\tan\frac{\beta}{2},\beta=2\tan^{-1}\left( \frac{k}{bv^2} \right)$$
Rutherfold (1911) fired $\alpha$ particles at gold leaf to obtain experimental results of the scattering.
But Scattering angles greater than $\pi/2$ is observed in the experiment, from which he concluded that the positive charge must be highly concentrated.


\documentclass{article}
\usepackage{amsmath,amssymb,amsthm,fullpage,enumerate,hyperref,graphicx}
\usepackage[color]{changebar}
\cbcolor{blue}
\theoremstyle{definition}
\newtheorem{defi}{Definition}
\newtheorem{theo}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{claim}{Claim}
\begin{document}
\section{Starting point}
We take a lattice
\[L=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2,\]
for some $\omega_1,\omega_2\in\mathbb{C}$ not on the same line. Then $\mathbb{C}/L$ is topologically a torus.

Since we're going to do differential geometry, we want to find an explicit map. We recall that the origin-centered torus is given parametrically as
\begin{align*}
  x(\theta,\phi)&=(R+r\cos(\theta))\cos(\phi)\\
  y(\theta,\phi)&=(R+r\cos(\theta))\sin(\phi)\\
  z(\theta,\phi)&=r\sin(\theta),
\end{align*}
where $0\leq\theta,\phi\leq 2\pi$ and $R$ is the distance from the center of the tube to the center of the torus, and $r$ is the radius of the tube. We denote this torus by $\mathbb{T}^2(R,r)$. If you plot $\mathbb{T}^2(R,r)$ you'll notice that $\theta$ specifies angle in relation to the center of the tube, and $\phi$ specifies angle in relation to the center of the torus. As in the below picture:

\begin{figure}[h]
  \centering
  \includegraphics[width=0.6\textwidth]{theta_phi.png}
\end{figure}

To figure out the map from $\mathbb{C}/L$ to $\mathbb{T}^2(R,r)$ we essentially just have to figure out how to map a point on the ``parallellogram with sides identified'' to a point on $\mathbb{T}^2(R,r)$ for some appropriate $R$ and $r$.

There are several ways of doing this, and here's but one. Let $F:\mathbb{C}\to\mathbb{T}^2(1,1)$ be given by
\[F(\alpha\omega_1+\beta\omega_2)=(x,y,z)(\alpha\cdot2\pi,\beta\cdot2\pi).\]
Since $\omega_1$ and $\omega_2$ are linearly independent, this is well-defined, and since $\cos$ and $\sin$ are $2\pi$-periodic, $F$ descends to a function $\tilde{F}:\mathbb{C}/L\to\mathbb{T}^2(R,r)$. It is clear that viewed as function from $\mathbb{R}^2$ to $\mathbb{R}^3$, $F$ is a diffeomorphism.

By the way, I'm making the gentle assumption that $\omega_1$ and $\omega_2$ are both in the first quadrant.

Excellent, so let's now bring in some definitions from Spivak so that we can make sense of $\mathrm{d}z$ -- the translation invariant one-form.
\section{Definitions from Spivak}
We begin with a $k$-dimensional manifold.
\begin{defi}
  A subset $M\subseteq\mathbb{R}^n$ is called a $k$-dimensional manifold if for
  every $x\in M$ it holds that
  \begin{quotation}
    there exists an open set $U\ni x$ and an open set $V\subseteq\mathbb{R}^n$ and
    a diffeomorphism $h:U\to V$ such that
    \[h(U\cap M)=V\cap(\mathbb{R}^k\times\{\mathbf{0}\}).\]
  \end{quotation}
\end{defi}
To see that $\mathbb{T}^2(R,r)$ is a $2$-manifold, we have to think a little bit. Take $v\in M$ arbitrary, say with $v=(x(\theta',\phi'),y(\theta',\phi),z(\theta',\phi'))$ with $0\leq\theta',\phi<2\pi$. Let $0\leq\epsilon<\pi/4$ be arbitrary and put
\[P=\{(x(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t),y(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t),z(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t)):-\epsilon<s,t<\epsilon\}.\]
Here $P$ stands for ``patch''. Let
\[w=(x(\pi/2,\phi'),y(\pi/2,\phi'),z(\pi/2,\phi')),\]
and let
\[U=\{p+rw:p\in P,-1/2<r<1/2\}.\]
Hence $U\cap M=P$. I take as geometrically clear that $U$ is indeed an open set. Let now
\[V=\{(s,t,u):-\epsilon<s,t<\epsilon,-1/2<u<1/2\},\]
and define $h:U\to V$ by
\[h(x(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t)+rw_1,y(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t)+rw_2,z(\theta'+2\pi\delta_{\theta',0}+s,\phi'+t)+rw_3)=(s,t,r).\]
Since any point in in $U$ is determined uniquely by the parameters $s$, $t$, and $r$, we have that $h$ is well-defined, and some first year calculus shows that it's smooth.

We see that
\[h(U\cap M)=h(P)=\{(s,t,0):-\epsilon<s,t<\epsilon\}=V\cap\mathbb{R}^2\times\{0\},\]
and so we conclude that $\mathbb{T}^2(R,r)$ indeed is a $2$-manifold.

I will take the view that $\mathbb{C}/L$ is simply identified with $\mathbb{T}^2(1,1)$ through $\tilde{F}$. This is because Spivak doesn't include an ambient space. {\tt Frankly, when giving the talk I will probably just use Lee's ``Smooth Manifolds'' instead or better yet Wells' ``Differential Analysis on Complex Manifolds'' because it's a bit difficult to translate notions all the time. On the other hand, Wells' and Lee kind of go off the deep end into some algebraic geometry. So there's that.}

Before continuing to {\it forms}, let's talk about coordinate systems. Spivak gives the following theorem.
\begin{theo}
  A subset $M$ of $\mathbb{R}^n$ is a $k$-dimensional manifold iff for every $x\in M$ the following condition is satisfied:
  \begin{quotation}
    There is an open set $U\ni x$ and an open set $W\subseteq\mathbb{R}^k$ and an injective smooth function $f:W\to\mathbb{R}^n$ such that
    \begin{enumerate}
      \item $f(W)=M\cap U$,
      \item $f'(y)$ has rank $k$ for every $y\in W$,
      \item $f^{-1}:f(W)\to W$ is continuous.
    \end{enumerate}
  \end{quotation}
  Such a function $f$ is called a coordinate system around $x$.
\end{theo}
I'm pretty sure that in modern terminology -- this corresponds to having an atlas.
\section{Forms}
Before we treat forms on manifolds, we need to recall what forms are in Euclidean space. For this, we need the concept of tensors, alternating tensors, and the wedge product. From Spivak's point of view, this isn't too bad. {\tt Although he seems to consciously skip the concept of vector and tangent bundles, which makes it a bit harder to define things.}

Let's start with tensors.
\begin{defi}[Tensor]
  Let $V$ be an $\mathbb{R}$-vector space, and let $k$ be a positive integer. Then a $k$-tensor on $V$ is a multilinear function $T:V^k\to\mathbb{R}$. The set of all $k$-tensors on $V$ is denoted by $\mathcal{T}^k(V)$. It forms a vector space given
  \[(S+T)(v)=S(v)+T(v),\]
  and
  \[(rS)(v)=rS(v),\]
  for $S,T\in\mathcal{T}^k(V)$ and $r\in\mathbb{R}$ and $v\in V$.
\end{defi}
In other words, $\mathcal{T}^k(V)=(V^k)^\vee$. If $T\in\mathcal{T}^k(V)$ it's called {\it alternating} if
\[T(v_1,\dots,v_i,\dots,v_j,\dots,v_k)=-T(v_1,\dots,v_j,\dots,v_i,\dots,v_k),\]
for any $i<j$. The space of all alternating $k$-tensors on $V$ is denoted by $\Lambda^k(V)$.

In this setting, the tensor product $\otimes$ is defined as follows.
\begin{defi}
  Let $V$ be an $\mathbb{R}$-vector space. Let $S\in\mathcal{T}^k(V)$ and $T\in\mathcal{T}^l(V)$. Then we define
  \[(S\otimes T)(v_1,\dots,v_k,v_{k+1},\dots,v_{k+l})=S(v_1,\dots,v_k)T(v_{k+1},\dots,v_{k+l}),\]
  and notice $S\otimes T\in\mathcal{T}^{k+l}(V)$.
\end{defi}
We can construct alternating tensors from tensors using the $\mathrm{Alt}$-map, defined as follows. Let $T\in\mathcal{T}^k(V)$, then put
\[\mathrm{Alt}(T)(v)=\frac{1}{k!}\sum_{\sigma\in S_k}\mathrm{sgn}(\sigma)T(\sigma.v),\]
where $\sigma$ acts by permutating the canonical basis. Clearly $\mathrm{Alt}$ is a generalization of the determinant.

It's a fact that if $T\in\mathcal{T}^k(V)$ then $\mathrm{Alt}(T)\in\Lambda^k(V)$ and $\mathrm{Alt}(\mathrm{Alt}(T))=T$, and if $\omega\in\Lambda^k(V)$, then $\mathrm{Alt}(\omega)=\omega$.

We can form products also on the space of alternating tensors.
\begin{defi}[Wedge product]
  Let $\omega\in\Lambda^k(V)$ and $\eta\in\Lambda^l(V)$ for some $\mathbb{R}$-vector space. Then we define
  \[\omega\land\eta=\frac{(k+l)!}{k!l!}\mathrm{Alt}(\omega\otimes\eta).\]
  It holds that $\omega\land\eta\in\Lambda^{k+l}(V)$.
\end{defi}
We can use the wedge product to get a basis for $\Lambda^k(V)$.
\begin{prop}
  Let $\{e_i\}_{i=1}^n$ be a basis for $V$. Then
  \[e_{i_1}^\vee\land\dots\land e_{i_k}^\vee,\]
  where $1\leq i_1<i_2<\dots<i_k\leq n$, forms a basis for $\Lambda^k(V)$.
\end{prop}

Now we're really close to defining forms. First we need the concept of a tangent space.
\begin{defi}[Tangent space]
  Let $n$ be a positive integer and let $p\in\mathbb{R}^n$. Then we define
  \[\mathbb{R}^n_p=\{(p,v):v\in\mathbb{R}^n\},\]
  and $(p,v)+(p,w)=(p,v+w)$ and $a(p,v)=(p,av)$. This turns $\mathbb{R}^n_p$ into a vector space, which we call the tangent space at $p$.

  We usually write $v_p$ for $(p,v)$.
\end{defi}
We will probably need the notion of vector fields, so I'll define that too.
\begin{defi}[Vector field]
  A vector field is a function
  \[F:\mathbb{R}^n\to\bigsqcup_{p\in\mathbb{R}^n}\mathbb{R}^n_p,\]
  such that $F(p)\in\mathbb{R}^n_p$ for every $p\in\mathbb{R}^n$.
\end{defi}
Notice that $(e_i)_p$ is a basis for $\mathbb{R}^n_p$, and thus we have
\[F(p)=\sum_{i=1}^nF^i(p)(e_i)_p,\]
for some component functions $F^i$. We call $F$ continuous or smooth if the $F^i$ are continuous or smooth; respectively.

And here we go, now we can define a differential form.
\begin{defi}[Differential form]
  A differential $k$-form on $\mathbb{R}^n$ is a function $\omega:\mathbb{R}^n\to\bigsqcup_{p\in\mathbb{R}^n}\Lambda^k(\mathbb{R}^n_p)$ with $\omega(p)\in\Lambda^k(\mathbb{R}^n_p)$.
\end{defi}
Since we know a basis for $\Lambda^k(V)$, we see that any $k$-form $\omega$ can be written as
\[\omega(p)=\sum_{1\leq i_1<\dots<i_k\leq n}\omega_{i_1,\dots,i_k}(p)\phi_{i_1}(p)\land\dots\land\phi_{i_k}(p),\]
for some coefficients $\omega_{i_1,\dots,i_k}(p)\in\mathbb{R}$, and $\phi_i=(e_i)_p^\vee$.

If the coefficient functions $p\mapsto\omega_{i_1,\dots,i_k}(p)$ are continuous or smooth, then we call the $k$-form $\omega$ continuous, or smooth; respectively.

Suppose now that $f:\mathbb{R}^n\to\mathbb{R}$ is smooth.
\begin{claim}
  Then $Df(p)\in\Lambda^1(\mathbb{R}^n)$.
\end{claim}
\begin{proof}
  By definition $Df(p)=\lambda:\mathbb{R}^n\to\mathbb{R}$ is the unique linear transformation satisfying
  \[\lim_{h\to 0}|f(p+h)-f(h)-\lambda(h)|/|h|=0.\]
  So $\lambda$, because it's linear, is a $1$-tensor. But then it's alternating (vacuously).
\end{proof}
Using this, we can define $df$, for $f:\mathbb{R}^n\to\mathbb{R}$ smooth.
\begin{defi}
  Let $f:\mathbb{R}^n\to\mathbb{R}$ be smooth. Then we define $df:\mathbb{R}^n\to\bigsqcup_{p\in\mathbb{R}^n}\Lambda^1(\mathbb{R}^n_p)$ by
  \[df(p)(v_p)=Df(p)(v).\]
\end{defi}
Let in particular $\pi^i:\mathbb{R}^n\to\mathbb{R}$ be defined by $\pi^i(x)=x^i$. Notice now that
\[d\pi^i(p)(v_p)=D\pi^i(p)(v)=e_i^Tv=v^i,\]
So $d\pi^i(p)((e_j)_p)=[i=j]$, so $d\pi^i(p)=(e_i)_p^\vee$. Usually we write $x^i$ for $\pi^i$, and thus any differential $k$-form can be written as
\[\omega(p)=\sum_{1\leq i_1<\dots<i_k\leq n}\omega_{i_1,\dots,i_k}(p)dx^{i_1}\land\dots\land dx^{i_k}.\]
When defining differential forms on manifolds, we want to induce smooth functions $\mathbb{R}^n\to\mathbb{R}^m$ to linear transformations $\mathbb{R}^n_p\to\mathbb{R}^n_{f(p)}$. This is how we do it.
\begin{defi}
  Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be a smooth. Then we define $f_\ast:\mathbb{R}^n_p\to\mathbb{R}^m_{f(p)}$ by
  \[f_\ast(v_p)=(Df(p)(v))_{f(p)}.\]
\end{defi}
\begin{claim}
  It makes sense.
\end{claim}
\begin{proof}
  We have that $Df(p)$ is a linear transformation $\mathbb{R}^n\to\mathbb{R}^m$, so $Df(p)(v)\in\mathbb{R}^m$ and thus $(Df(p)(v))_{f(p)}$. Linearity should be fine.
\end{proof}
From $f_\ast$ we can induce further, to get a transformation
\[f^\ast:\Lambda^k(\mathbb{R}^m_{f(p)})\to\Lambda^k(\mathbb{R}^n_p),\]
in the usual way -- that is:
\[f^\ast(\omega)=\omega\circ f_\ast.\]
\begin{claim}
  This also makes sense.
\end{claim}
\begin{proof}
  Take $v_p\in\mathbb{R}^n_p$, then $f_\ast(v_p)\in\mathbb{R}^m_{f(p)}$, so is in the domain of $\omega$. Cool, we're fine.
\end{proof}
That's all we need for forms. So let's go to {\Huge\tt manifold town!}
\section{Forms on manifolds}
Let $M$ be a $k$-dimensional in $\mathbb{R}^n$ and let $f:W\to\mathbb{R}^n$ be a coordinate system around $x=f(a)$.
\begin{claim}
  It holds that $f_\ast:\mathbb{R}^k_a\to\mathbb{R}^n_x$ is injective, and $f_\ast(\mathbb{R}^k_a)$ is $k$-dimensional subspace of $\mathbb{R}^n_x$.
\end{claim}
\begin{proof}
  We have that $f_\ast(v_a)=(Df(a)(v))_x$. Since $f'(a)$ has rank $k$, the linear transformation $Df(a):W\to\mathbb{R}^k$ is an isomorphism. Therefore, $(Df(a)(v))_x=0$ implies $Df(a)(v)=0$ implies $v=0$ implies $v_a=0$. So also $f_\ast$ is injective.

  It's not surjective, but at least $f_\ast(\mathbb{R}^k_a)$ is an isomorphic copy of $\mathbb{R}^k_a$, so it is of dimension $k$.
\end{proof}
Let $g:V\to\mathbb{R}^n$ be any other coordinate system around $x=g(b)$. {\tt We assume, I think WLOG, that the $U\ni x$ is the same for both.}
\begin{claim}
  Its range is the same.
\end{claim}
\begin{proof}
  \cbstart
  Let's first prove that $g_\ast=f_\ast\circ(f^{-1}\circ g)_\ast$. This follows basically by the chain rule. Note that
  \[f^{-1}(g(b))=f^{-1}(x)=a,\]
  and therefore
  \[Df(a)D(f^{-1}\circ g)(b)=Df(f^{-1}(g(b)))D(f^{-1}\circ g)(b)=D(f\circ(f^{-1}\circ g))(b)=Dg(b).\]
  This means that
  \[f_\ast(f^{-1}\circ g)_\ast(v_b)=f_\ast((D(f^{-1}\circ g)(b)v)_a)=(Df(a)(D(f^{-1}\circ g)(b)v))_x=(Dg(b)v)_x=g_\ast(v_b).\]
  Let's now take a peek at
  \[(f^{-1}\circ g)_\ast(\mathbb{R}^k_b).\]
  We have that
  \[(f^{-1}\circ g)_\ast(\mathbb{R}^k_b)=\{(Df^{-1}g(b)\circ Dg(b)v)_a:v\in\mathbb{R}^k\}.\]
  Since $Dg(b)$ has rank $k$, we have that $Dg(b)v$ spans a $k$-dimensional subspace of $\mathbb{R}^n$. Since $Df(a)$ has rank $k$, it should be the case that $Df^{-1}(g(b))$ (which exists because reasons) also has rank $k$, and thus $Df^{-1}(g(b))Dg(b)v$ should be $k$-dimensional, and thus $\mathbb{R}^k$ itself.
  \cbend

  {\tt BTW:} The correct way is to say that $(g^{-1}\circ f)_\ast$ is an inverse to $(f^{-1}\circ g)_\ast$. No smoothness needed!
\end{proof}
{\tt Fishy argument. I need to fix this.}

The best way to fix this is to just use Lee. But, I'll let it slide for now. Let's just assume that it works. {\tt Damn it!}

{\tt Found a fix in ``Comprehensive Introduction'' volume 1. Yes!}

If $f:W\to\mathbb{R}^n$ is a coordinate system. We write
\[M_x=f_\ast(\mathbb{R}^k_a),\]
and call this the tangent space of $M$ at $x$. We can now define a differential $p$-form.
\begin{defi}[Form on a manifold]
  Let $\omega:M\to\bigsqcup_{x\in M}\Lambda^p(M_x)$ satisfy $\omega(x)\in\Lambda^p(M_x)$ for every $x\in M$. Then we call $\omega$ a $p$-form on $M$.

  If $f:W\to\mathbb{R}^n$ is a coordinate system, then $f^\ast\omega$ is a $p$-form on $W$, and we say that $\omega$ is smooth if $f^\ast\omega$ is.
\end{defi}
Next Spivak claims we can write $\omega$ as
\[\omega=\sum_{1\leq i_1<\dots<i_p\leq n}\omega_{i_1,\dots,i_p}dx^{i_1}\land\dots\land dx^{i_p},\]
and this means we need to make sense of
\[dx^{i_j}(p),\]
for $p\in M$. Can we? Previously we defined it as
\[d\pi^i(v)(w\in M_x)=D\pi^i(v)(w),\]
but we probably need to use coordinate systems now.

And yes we do, but recall that $M_x=f_\ast(\mathbb{R}^k_a)$ actually sits inside $\mathbb{R}^n_x$, so we in fact use the same $d\pi^i$ as before.

But, when we want to carry over $dz$ to $\mathbb{T}^2$ we get a problem. It's the case that
\[f_\ast(\mathbb{R}^2_a)\subset\mathbb{R}^3_x,\]
so we can impossible make sense of $(dx,dy)$, because $dx$ and $dy$ only make sense in $\mathbb{R}^2_x$. Here's where our grand plan stops making sense.
\end{document}