+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Category theory +--{: .hide} [[!include category theory - contents]] =-- #### Limits and colimits +--{: .hide} [[!include infinity-limits - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The **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]] **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]]. ## 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 ($M$)-**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 **[[corestriction]]** of $f$ (or **coastriction**, see [mathoverflow](http://mathoverflow.net/questions/29911/whats-the-notation-for-a-function-restricted-to-a-subset-of-the-codomain/65813#65813)): 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 _[[regular image]]_. See [below](#AsEqualizer) 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]] $$C/d \to M/d$$ 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. ## Examples * In [[Set]], for $M=$ monomorphisms = [[injections]], this reproduces the ordinary notion of image. * More generally in any [[topos]] the [[(epi,mono) factorization system]] gives the factorization through the image $f \colon A \overset{epi}{\to} im(f) \overset{mono}{\to} B$. * Similarly in an [[abelian category]] ([by definition](abelian+category#AbelianCategory)) [[(epi,mono) factorization system|(epi,mono) factorization]] gives factorization through the image. * In algebraic categories such as [[Grp]], for $M=$ monomorphisms, this also reproduces the ordinary notions of image. * In [[Top]], for $M=$ [[subspace]] inclusions, the $M$-image is the set-theoretic image topologized as a [[subspace]] of the [[target|codomain]], see [[subspace topology]] and more comprehensive [Introduction to topology](https://ncatlab.org/nlab/show/Introduction+to+Topology+--+1#subspaces). 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]]. * 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]]. * In [[Cat]] (considered as a [[1-category]] of [[strict categories]]), 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 $A$. {#RepleteImages} See also at at *[[replete subcategory]]* the section *[Repletion and replete images](replete+subcategory#RepletionAndRepleteImages)*. Often it is desireable to instead treat $Cat$ as a [[2-category]], in which case one can use a more 2-categorical notion of image. See, for instance, *[[full image]]* and *[[essential image]]*. ## Properties ### Basic properties +-- {: .num_prop #PreservationOfUnionsAndIntersectionsOfSets} ###### Proposition **([[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. 1. $im_f\left( \underset{i \in I}{\cup} S_i\right) = \left(\underset{i \in I}{\cup} im_f(S_i)\right)$ 1. $im_f\left( \underset{i \in I}{\cap} S_i\right) \subset \left(\underset{i \in I}{\cap} im_f(S_i)\right)$ 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]]: * $(f\,\text{injective}) \Rightarrow \left(im_f\left( \underset{i \in I}{\cap} S_i\right) = \left(\underset{i \in I}{\cap} im_f(S_i)\right)\right)$ =-- For details see at _[[interactions of images and pre-images with unions and intersections]]_. ### 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 $$\array{a & \overset{h}{\to} & b\\ ^e\downarrow && \downarrow^n\\ c & \underset{k}{\to} & d}$$ 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. Cf. at *[[(epi, mono) factorization system]]*. 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. **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 **$E$-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." ### Construction via limits {#AsEqualizer} Suppose that the category $C$ admits [[finite limits]] and [[finite 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 $$ Im f \simeq lim (d \rightrightarrows d \sqcup_c d) \,, $$ where $d \sqcup_c d$ denotes the [[pushout]] $$ \array{ c &\stackrel{f}{\to}& d \\ \downarrow^{f} && \downarrow \\ d &\to& d \sqcup_c d } \,. $$ In other words, the **[[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 **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 **image** and **coimage** respectively. This can be confusing, however, since in many cases (such as in any [[regular category]]) the *regular coimage* coincides with the $M$-image for $M=Mono$ the class of all monomorphisms, which it is also natural to simply call the *image*. ### 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: +-- {: .num_lemma} ###### Lemma If $C$ has finite limits and colimits, then there is a unique map $$ u : coim f \to im f $$ from its regular coimage to its regular image such that $$ f = (c \to coim f \stackrel{u}{\to} im f \to d) \,. $$ =-- +-- {: .proof} ###### Proof Because $f$ coequalizes $c \times_d c \rightrightarrows c$, a morphism $h$ in $$ \array{ c &\times_d c \rightrightarrows& c &\stackrel{f}{\to}& d &\rightrightarrows& d \sqcup_{c} d \\ && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono} \\ && coim f && im f } $$ 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 $$ \array{ c &\times_d c \rightrightarrows& c &\stackrel{f}{\to}& d &\rightrightarrows& d \sqcup_{c} d \\ && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono} \\ && coim f &\stackrel{u}{\to}& im f } $$ exists uniquely. =-- 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. ## In higher category theory In [[higher category theory]] there are generalizations of the notion of image, such as these: * [[essential image]] (of a [[functor]]) * [[homotopy image]]. 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 **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]]. ### In $(\infty,1)$-category theory {#InfImage} A **(regular) $(\infty,1)$-image** of a morphism $f : c \to d$ in an [[(∞,1)-category]] with [[(∞,1)-limit|(∞,1)-limits]] and -colimits should be defined to be the [[(∞,1)-limit]] over the [[Čech nerve|Cech co-nerve]] of $f$: $$ im f := \lim_{\leftarrow} \left( d \rightrightarrows d \coprod_c d \stackrel{\to}{\rightrightarrows} d \coprod_c d \coprod_c d \stackrel{\to}{\stackrel{\to}{\rightrightarrows}} \cdots \right) \,. $$ Notice that * this reduces to the above [equalizer definition](#AsEqualizer) in the case that the ambient $(\infty,1)$-category is just an ordinary category; * this implies that the inclusion $im f \to d$ is a [[regular monomorphism]] in the $(\infty,1)$-category sense (described [here](http://ncatlab.org/nlab/show/regular+monomorphism#Infty1Version)). For more see _[[n-image]]_. #### Examples {#InfImageExamples} Applied to the $(\infty,1)$-category [[∞Grpd]] this gives a notion of image of [[(∞,1)-functor|(∞,1)-functors]] between [[∞-groupoid|∞-groupoids]] and hence a notion of image of [[functor|functors]] between [[groupoid|groupoids]], [[2-functor|2-functors]] between [[2-groupoid|2-groupoids]], etc. ## Related concepts * [[preimage]] * [[coimage]] * [[inverse image]], [[direct image]] * [[kernel]], [[cokernel]] * [[continuous images of compact spaces are compact]] * [[value]] [[!redirects images]] [[!redirects regular image]] [[!redirects regular images]] [[!redirects regular coimage]] [[!redirects regular coimages]] [[!redirects image factorization]] [[!redirects image factorizations]]