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\begin{document}

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\section*{image}

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

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

[[!include category theory - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{relation_to_factorization_systems}{Relation to factorization systems}\dotfill \pageref*{relation_to_factorization_systems} \linebreak
\noindent\hyperlink{AsEqualizer}{Construction via limits}\dotfill \pageref*{AsEqualizer} \linebreak
\noindent\hyperlink{comparison_of_regular_images_and_coimages}{Comparison of regular images and coimages}\dotfill \pageref*{comparison_of_regular_images_and_coimages} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{InfImage}{In $(\infty,1)$-category theory}\dotfill \pageref*{InfImage} \linebreak
\noindent\hyperlink{InfImageExamples}{Examples}\dotfill \pageref*{InfImageExamples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \textbf{image} of a [[function]] $f\colon A\to B$ between [[sets]] is the [[subset]] of $B$ consisting of all those elements $b\in B$ that are of the form $f(a)$ for some $a\in A$. This notion can be generalized from [[Set]] to other categories, as follows.

To discuss images in a category $C$, we must first fix a notion of [[subobject]] or [[embedding]] in $C$. (Sometimes we want these to be the [[monomorphisms]], but sometimes we want the [[regular monomorphisms]] instead.) Then [[generalized the|the]] \textbf{image} of a [[morphism]] $f\colon A\to B$ in $C$ is a universal factorization of $f$ into a composite $A \to im(f) \to B$ such that $im(f)\to B$ is a subobject, of the specified sort.

Note that in this generality, a given morphism may or may not have an image, although if it does, it is unique up to isomorphism by universality. In many cases, images can be constructed out of [[limits]] and [[colimits]] in the ambient category. In particular, in a [[regular category]], images (relative to all monomorphisms) can be constructed as the [[quotient object]] of a [[kernel pair]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[category]], let $M\subset Mono(C)$ be a subclass of the monomorphisms in $C$, and let $f: c \to d$ be a [[morphism]] in $C$. The ($M$)-\textbf{image} of $f$ is the smallest $M$-[[subobject]] $im(f) \hookrightarrow d$ through which $f$ factors (if it exists). The factorizing morphism $c \to im(f)$ is sometimes called the \textbf{corestriction} of $f$ (or \textbf{coastriction}, see \href{http://mathoverflow.net/questions/29911/whats-the-notation-for-a-function-restricted-to-a-subset-of-the-codomain/65813#65813}{mathoverflow}):

In other words, it is a factorization $c \overset{e}{\to} im(f) \overset{m}{\to} d$ of $f$ (i.e. $f = m e$) such that $m\in M$, and given any other factorization $f = m' e'$ with $m'\in M$, we have $m \subseteq m'$ as subobjects of $C$ (i.e. $m$ factors through $m'$, $m = m' k$ for some $k$). Such a factorization is unique up to unique isomorphism, if it exists.

(For instance if $M$ is the class of [[regular monomorphisms]], then the $M$-image is the \emph{[[regular image]]}. See \hyperlink{AsEqualizer}{below} for more.)

This can be phrased equivalently as follows. Let $C/d$ be the [[slice category]] of $C$ over $d$, and let $M/d$ be its full subcategory whose objects are $M$-morphisms into $d$. If all images exist in $C$, then taking the image of a map $f: c \to d$ provides a [[left adjoint]]

\begin{displaymath}
C/d \to M/d
\end{displaymath}
to the inclusion $M/d \hookrightarrow C/d$. More generally, an image of a single morphism $f\colon c\to d$ is a [[universal arrow]] from $f$ to this inclusion.

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

\begin{itemize}%
\item In [[Set]], for $M=$ monomorphisms = [[injections]], this reproduces the ordinary notion of image.


\item More generally in any [[topos]] the [[(epi,mono) factorization system]] gives the factorization through the image $f \colon A \overset{epi}{\to} im(f) \overset{mono}{\to} B$.


\item Similarly in an [[abelian category]] (\href{abelian+category#AbelianCategory}{by definition}) [[(epi,mono) factorization system|(epi,mono) factorization]] gives factorization through the image.


\item In algebraic categories such as [[Grp]], for $M=$ monomorphisms, this also reproduces the ordinary notions of image.


\item In [[Top]], for $M=$ [[subspace]] inclusions, the $M$-image is the set-theoretic image topologized as a [[subspace]] of the [[target|codomain]]. On the other hand, for $M=$ injective continuous maps, the $M$-image is the set-theoretic image topologized as a [[quotient space]] of the [[source|domain]]. For more basic details see at \emph{[[Introduction to Topology -- 1]]} \hyperlink{ImageFactorization}{here}.


\item A [[regular category]] can be defined as a [[finitely complete category]] in which all images exist for $M=$ monomorphisms, and such images are moreover stable under [[pullback]]. In particular, this includes any [[topos]].


\item In [[Cat]] (considered as a [[1-category]]), the image of a [[functor]] $F\colon A\to B$ is the smallest [[subcategory]] of $B$ which contains images through $F$ of all morphisms in $A$. Note that some of the morphisms in the image may not be images of any morphism in $A$; all morphisms in the image of $F$ are [[composite|compositions]] in $B$ of $B$-composable sequences of images of morphisms in $A$, but these themselves do not necessarily form $A$-composable sequences of morphisms in $A$.

Usually it is better to treat $Cat$ as a 2-category, in which case one can use a more 2-categorical notion of image. See, for instance, [[full image]], [[essential image]], and [[replete image]].



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{prop}
\label{PreservationOfUnionsAndIntersectionsOfSets}\hypertarget{PreservationOfUnionsAndIntersectionsOfSets}{}
\textbf{([[images preserve unions]] but not in general [[intersections]])}

Let $f \colon X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $\{S_i \subset X\}_{i \in I}$ be a set of [[subsets]] of $X$. Then.

\begin{enumerate}%
\item $im_f\left( \underset{i \in I}{\cup}  S_i\right) = \left(\underset{i \in I}{\cup} im_f(S_i)\right)$


\item $im_f\left( \underset{i \in I}{\cap}  S_i\right) \subset \left(\underset{i \in I}{\cap} im_f(S_i)\right)$



\end{enumerate}
The inclusion in the second item is in general proper. If $f$ is an [[injective function]] and $I$ is [[inhabited set|inhabited]] then this is a [[bijection]]:

\begin{itemize}%
\item $(f\,\text{injective}) \Rightarrow \left(im_f\left( \underset{i \in I}{\cap}  S_i\right) = \left(\underset{i \in I}{\cap} im_f(S_i)\right)\right)$

\end{itemize}
\end{prop}
For details see at \emph{[[interactions of images and pre-images with unions and intersections]]}.

\hypertarget{relation_to_factorization_systems}{}\subsubsection*{{Relation to factorization systems}}\label{relation_to_factorization_systems}

Suppose that $M$ is closed under composition, and that $f = m e$ is an image factorization relative to $M$. Then $e$ has the property that if $e = n g$ for some $n\in M$, then $n$ is an isomorphism --- for then we would have $f = (m n) g$ and so by universality of images, $m$ would factor through $m n$. In particular, if $M$ is the class of all monomorphisms and $C$ has [[equalizers]], then $e$ is an [[extremal epimorphism]].

If $C$ has [[pullbacks]] and $M$ is closed under pullbacks, then we can say more: $e$ is [[orthogonality|orthogonal]] to $M$. For if

\begin{displaymath}
\itexarray{a & \overset{h}{\to} & b\\
  ^e\downarrow && \downarrow^n\\
  c  & \underset{k}{\to} & d}
\end{displaymath}
is a commutative square with $n\in M$, then the pullback $k^*n$ is an $M$-morphism through which $e$ factors. Hence $k^*n$ must be an isomorphism, and so the square admits a diagonal filler, which is unique since $n\in M$ is monic. It follows that if all $M$-images exist in $C$, then $M$ is the right class of an [[orthogonal factorization system]], and $M$-images are precisely the factorizations in this OFS.

Conversely, it is easy to see that if $(E,M)$ is an OFS on a category $C$, then all $M$-images exist and are given by the factorizations of the OFS. Therefore, to give a notion of image is more or less equivalent to giving an orthogonal factorization system.

\textbf{Duality}

Note that the notion of factorization system is self-dual. Therefore, if $(E,M)$ is a factorization system and $c \overset{e}{\to} a \overset{m}{\to} d$ is an $(E,M)$-factorization of $f\colon c\to d$, then not only is $m$ the $M$-image of $f$ (the least $M$-subobject through which $f$ factors), but dually $e$ is also the \textbf{$E$-coimage} of $f$, i.e. the greatest $E$-quotient through which $f$ factors.

However, see below for additional remarks on the usage of the terms ``image'' and ``coimage.''

\hypertarget{AsEqualizer}{}\subsubsection*{{Construction via limits}}\label{AsEqualizer}

Suppose that the category $C$ admits finite [[limits]] and [[colimits]], and that $M=RegMono$ consists of the [[regular monomorphisms]]. Then the $M$-image of a morphism $f : c \to d$ may be constructed as

\begin{displaymath}
Im f \simeq lim (d \rightrightarrows d \sqcup_c d)
  \,,
\end{displaymath}
where $d \sqcup_c d$ denotes the [[pushout]]

\begin{displaymath}
\itexarray{
    c &\stackrel{f}{\to}& d
    \\
    \downarrow^{f} && \downarrow
    \\
    d &\to& d \sqcup_c d
  }
  \,.
\end{displaymath}
In other words, the \textbf{[[regular image]]} is the [[equalizer]] of the [[cokernel pair]]. To see that this is in fact the $RegMono$-image, we first note that it is of course a regular monomorphism by definition, and then invoke the fact that in a category with [[finite limits]] and colimits, a monomorphism is regular if and only if it is the equalizer of its cokernel pair.

Dually, the \textbf{regular coimage} of a morphism is the [[coequalizer]] of its [[kernel pair]]. In [[Set]] (and more generally in any [[topos]]) these two constructions coincide, but in general they are distinct. For example, in [[Top]] the regular image is the set-theoretic image topologized as a subspace of the [[target|codomain]], while the regular coimage is the set-theoretic image topologized as a quotient space of the [[source|domain]].

Note that some authors drop the ``regular'' and simply call these constructions the \textbf{image} and \textbf{coimage} respectively. This can be confusing, however, since in many cases (such as in any [[regular category]]) the \emph{regular coimage} coincides with the $M$-image for $M=Mono$ the class of all monomorphisms, which it is also natural to simply call the \emph{image}.

\hypertarget{comparison_of_regular_images_and_coimages}{}\subsubsection*{{Comparison of regular images and coimages}}\label{comparison_of_regular_images_and_coimages}

Suppose that $M_1$ and $M_2$ are two classes with $M_1\subseteq M_2$. If $f$ has both an $M_1$-image $im_1(f)$ and an $M_2$-image $im_2(f)$, then by universality, the latter must factor through the former. The correspondence between images and factorization systems also extends to pairs; see [[ternary factorization system]].

As a special case of this, we have:

\begin{lemma}
\label{}\hypertarget{}{}
If $C$ has finite limits and colimits, then there is a unique map

\begin{displaymath}
u :  coim f \to im f
\end{displaymath}
from its regular coimage to its regular image such that

\begin{displaymath}
f = (c \to coim f \stackrel{u}{\to} im f \to d)
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
Because $f$ coequalizes $c \times_d c \rightrightarrows c$, a morphism $h$ in

\begin{displaymath}
\itexarray{
    c 
      &\times_d c \rightrightarrows&
    c
    &\stackrel{f}{\to}&
    d
    &\rightrightarrows&
    d \sqcup_{c} d
    \\
   && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono}
   \\
    && coim f && im f
  }
\end{displaymath}
exists uniquely.

Because $c \to coim f$ is [[epimorphism|epi]] it follows that $h$ equalizes $d \rightrightarrows d \sqcup_c d$ and hence $u$ in the diagram

\begin{displaymath}
\itexarray{
    c 
      &\times_d c \rightrightarrows&
    c
    &\stackrel{f}{\to}&
    d
    &\rightrightarrows&
    d \sqcup_{c} d
    \\
   && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono}
   \\
    && coim f &\stackrel{u}{\to}& im f
  }
\end{displaymath}
exists uniquely.

\end{proof}
If this map $u$ is an [[isomorphism]], then $f$ is sometimes called a [[strict morphism]]. In particular, if $C$ has finite limits and colimits and every morphism is a strict morphism, then the regular image and regular coimage factorizations coincide and give an epi-mono factorization system.

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

In [[higher category theory]] there are generalizations of the notion of image, such as these:

\begin{itemize}%
\item [[essential image]] (of a [[functor]])
\item [[homotopy image]].

\end{itemize}
However, it is not clear that either serves as the proper categorification of the notion described above.

There are several properties we might want a `higher image' to have. For example, in an $2$-category, we might want isomorphic 1-cells to have equivalent images. In \textbf{Cat}, we might want the image of a functor between discrete categories to be its image as a function. One fruitful direction is to study a [[factorization system in a 2-category]].

\hypertarget{InfImage}{}\subsubsection*{{In $(\infty,1)$-category theory}}\label{InfImage}

A \textbf{(regular) $(\infty,1)$-image} of a morphism $f : c \to d$ in an [[(∞,1)-category]] with [[(∞,1)-limit]]s and -colimits should be defined to be the [[(∞,1)-limit]] over the [[Cech nerve|Cech co-nerve]] of $f$:

\begin{displaymath}
im f := \lim_{\leftarrow}
  \left(
    d \rightrightarrows d \coprod_c d 
    \stackrel{\to}{\rightrightarrows}
    d \coprod_c d \coprod_c d
    \stackrel{\to}{\stackrel{\to}{\rightrightarrows}}
    \cdots
  \right)
  \,.
\end{displaymath}
Notice that

\begin{itemize}%
\item this reduces to the above \hyperlink{AsEqualizer}{equalizer definition} in the case that the ambient $(\infty,1)$-category is just an ordinary category;


\item this implies that the inclusion $im f \to d$ is a [[regular monomorphism]] in the $(\infty,1)$-category sense (described \href{http://ncatlab.org/nlab/show/regular+monomorphism#Infty1Version}{here}).



\end{itemize}
For more see \emph{[[n-image]]}.

\hypertarget{InfImageExamples}{}\paragraph*{{Examples}}\label{InfImageExamples}

Applied to the $(\infty,1)$-category [[∞Grpd]] this gives a notion of image of [[(∞,1)-functor]]s between [[∞-groupoid]]s and hence a notion of image of [[functor]]s between [[groupoid]]s, [[2-functor]]s between [[2-groupoid]]s, etc.

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

\begin{itemize}%
\item [[preimage]]


\item [[coimage]]


\item [[inverse image]], [[direct image]]


\item [[kernel]], [[cokernel]]


\item [[continuous images of compact spaces are compact]]



\end{itemize}
[[!redirects images]] [[!redirects corestriction]] [[!redirects regular image]] [[!redirects regular coimage]]

[[!redirects image factorization]] [[!redirects image factorizations]]



\end{document}