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\begin{document}

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\section*{embedding of categories}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{1categorical_definitions}{1-categorical definitions}\dotfill \pageref*{1categorical_definitions} \linebreak
\noindent\hyperlink{2categorical_definitions}{2-categorical definitions}\dotfill \pageref*{2categorical_definitions} \linebreak
\noindent\hyperlink{relationship}{Relationship}\dotfill \pageref*{relationship} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_yoneda_embedding}{The Yoneda embedding}\dotfill \pageref*{the_yoneda_embedding} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In much of mathematics, certainly in the traditional treatment of mathematical [[structure in model theory|structures]] from the point of view of [[logic]] and [[model theory]], an ``[[embedding]]'' is typically a [[monomorphism]] between structures that preserves \emph{and reflects} the structure specified within a given [[language]] or [[theory]]. Such a notion is often definable directly in terms of [[category theory]], i.e., in terms of properties of morphisms in the designated category of structures and structure-preserving maps.

But for categories themselves, the situation and practice is more complicated, with no notion of ``embedding of categories'' that is universally accepted in the literature. A critical dividing line, leading to several distinct notions of embedding, lies in how one views the [[Cat|category of categories]]:

\begin{itemize}%
\item If we view categories as mathematical structures in the traditional sense, with central concepts being definable in the ordinary 1-category of categories and [[functors]] (which are the relevant structure-preserving maps), then notions of embedding tend to emphasize they should be [[injective maps]] on the classes of objects (and more besides).


\item If we view categories however as having a higher-dimensional aspect, i.e., if we view central concepts as being based rather on the [[2-category]] of categories, functors, \emph{and} [[natural transformations]], then different notions of embedding come to the fore, where notably the assumption of injectivity on the object-classes clashes with the [[principle of equivalence]] (and is therefore rejected).



\end{itemize}
In this article we survey the various notions of categorical embedding that have appeared in the literature, and try to describe some of the underlying contexts and rationales.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{1categorical_definitions}{}\subsubsection*{{1-categorical definitions}}\label{1categorical_definitions}

Perhaps the most obvious 1-categorical definition of embedding is:

\begin{itemize}%
\item a [[functor]] $F: C \to D$ that is both injective on objects and a [[faithful functor]].

\end{itemize}
This definition is adopted in \hyperlink{AHS}{Adamek-Herrlich-Strecker} and in \hyperlink{Riehl}{Riehl}, for example. In this definition we accept that categories are [[strict categories]], where objects can be compared for [[equality]]. Embeddings in this sense are straightforwardly the same thing as [[monomorphisms]] in the $1$-category [[Cat]].

If we define categories with \href{/nlab/show/category#OneCollectionOfMorphisms}{one collection of morphisms}, then this definition is equivalent to saying that the action of $F$ on morphisms is an [[injective function]] $Mor(C) \to Mor(D)$. In particular, if this latter condition holds and $F(c) = F(c')$ for objects $c, c'$ of $C$, then $F(1_c) = F(1_{c'})$ and thus $1_c = 1_{c'}$ by injectivity on morphisms, whence $c = c'$.

However, as noted above, in many cases an ``embedding'' means something \emph{stronger} than a mere monomorphism, being required to reflect ``all the structure''. For instance, an embedding of [[topological spaces]] is a subspace inclusion, not merely an injective continuous map (the latter being the monos in [[Top]]).

In particular, the above definition of embedding does not even reflect the property of objects being isomorphic, so that it is not necessarily injective on isomorphism classes of objects! This suggests that a (still 1-categorical) embedding of categories ought to be something stronger. Some possibilities include:

\begin{itemize}%
\item The [[regular monomorphisms]] or [[extremal monomorphisms]] in [[Cat]] (what are those?)
\item Monomorphisms in $Cat$ that are also [[full functors]], hence [[fully faithful functors]]. These are sometimes called \textbf{full embeddings}.
\item Monomorphisms in $Cat$ that are also injective on isomorphism classes of objects (this is used in [[Toposes, Triples, and Theories]]), or more strongly that are also [[pseudomonic functors]].

\end{itemize}
\hypertarget{2categorical_definitions}{}\subsubsection*{{2-categorical definitions}}\label{2categorical_definitions}

Most of the above 1-categorical notions of embedding corresponds to a ``pure'' 2-categorical notion by simply removing the injectivity on objects. Thus we have the following candidates:

\begin{itemize}%
\item [[faithful functors]]
\item [[fully faithful functors]]
\item [[pseudomonic functors]]

\end{itemize}
The choice between these is closely related to the question of giving an equivalence-invariant notion of [[subcategory]].

\hypertarget{relationship}{}\subsubsection*{{Relationship}}\label{relationship}

Every fully faithful functor is \emph{equivalent} to one that is fully faithful and injective on objects. Let $f:C\to D$ be fully faithful, and let $E$ be the ``mapping cylinder'' category, whose objects are the disjoint unions of those of $C$ and $D$, with $C$ and $D$ embedded fully-faithfully (and of course injectively on objects), and $E(c,d) = D(f c,d)$ and $E(d,c) = D(d,f c)$. Full-faithfulness of $f$ allow us to compose all arrows in $E$ making it a category. There is a projection $E\to D$ that is the identity on $D$ and acts by $f$ on $C$, and which is an inverse equivalence to the inclusion $D\to E$. Thus, $f:C\to D$ is equivalent, in the 2-category whose objects are functors and whose morphisms are pseudo-commutative squares, to the inclusion functor $C\to E$, which is fully faithful and injective on objects.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{the_yoneda_embedding}{}\subsubsection*{{The Yoneda embedding}}\label{the_yoneda_embedding}

Probably the most common embedding of categories encountered is the [[Yoneda embedding]]. This is fully faithful, but whether or not it is always injective on objects depends on set-theoretic details of how we define categories. If we use a definition of category with \href{/nlab/show/category#AFamilyOfCollectionsOfMorphisms}{a family of collections of morphisms}, then it might happen that two distinct (but necessarily isomorphic) objects $x$ and $y$ have $C(z,x) = C(z,y)$ as sets for all $z\in C$, so that the Yoneda embedding would take $x$ and $y$ to literally the same presheaf. (Such an equality of hom-sets is not as weird as it might sound; for instance, when regarding a [[preordered set]] as a category it's natural to define every nonempty homset to be the \emph{same} [[terminal set]].)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Barr embedding theorem]]


\item [[embedding]]


\item [[subcategory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Jiří Adámek]], [[Horst Herrlich]], and [[George Strecker]], \emph{Abstract and Concrete Categories: The Joy of Cats}, John Wiley and Sons 1990. (\href{http://katmat.math.uni-bremen.de/acc/}{Online edition})


\item [[Emily Riehl]], \emph{Category Theory in Context}, Dover Publications, Inc. 2016. (\href{http://www.math.jhu.edu/~eriehl/context.pdf}{pdf})



\end{itemize}
[[!redirects embedding of categories]]



\end{document}