nLab
image

Context

Category theory

Limits and colimits

Contents

Idea

The image of a function f:ABf\colon A\to B between sets is the subset of BB consisting of all those elements bBb\in B that are of the form f(a)f(a) for some aAa\in A. This notion can be generalized from Set to other categories, as follows.

To discuss images in a category CC, we must first fix a notion of subobject or embedding in CC. (Sometimes we want these to be the monomorphisms, but sometimes we want the regular monomorphisms instead.) Then the image of a morphism f:ABf\colon A\to B in CC is a universal factorization of ff into a composite Aim(f)BA \to im(f) \to B such that im(f)Bim(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 CC be a category, let MMono(C)M\subset Mono(C) be a subclass of the monomorphisms in CC, and let f:cdf: c \to d be a morphism in CC. The image of ff is the smallest MM-subobject im(f)dim(f) \hookrightarrow d through which ff factors (if it exists). The factorizing morphism cim(f)c \to im(f) is sometimes called the corestriction of ff (or coastriction, see mathoverflow):

In other words, it is a factorization ceim(f)mdc \overset{e}{\to} im(f) \overset{m}{\to} d of ff (i.e. f=mef = m e) such that mMm\in M, and given any other factorization f=mef = m' e' with mMm'\in M, we have mmm \subseteq m' as subobjects of CC (i.e. mm factors through mm', m=mkm = m' k for some kk). Such a factorization is unique up to unique isomorphism, if it exists.

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

C/dM/dC/d \to M/d

to the inclusion M/dC/dM/d \hookrightarrow C/d. More generally, an image of a single morphism f:cdf\colon c\to d is a universal arrow from ff to this inclusion.

Examples

  • In Set, for M=M= monomorphisms = injections, this reproduces the ordinary notion of image.

  • In algebraic categories such as Grp, for M=M= monomorphisms, this also reproduces the ordinary notions of image.

  • In Top, for M=M= subspace inclusions, the MM-image is the set-theoretic image topologized as a subspace of the codomain. On the other hand, for M=M= injective continuous maps, the MM-image is the set-theoretic image topologized as a quotient space of the domain.

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

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

    Usually it is better to treat CatCat 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.

Relation to factorization systems

Suppose that MM is closed under composition, and that f=mef = m e is an image factorization relative to MM. Then ee has the property that if e=nge = n g for some nMn\in M, then nn is an isomorphism — for then we would have f=(mn)gf = (m n) g and so by universality of images, mm would factor through mnm n. In particular, if MM is the class of all monomorphisms and CC has equalizers, then ee is an extremal epimorphism.

If CC has pullbacks and MM is closed under pullbacks, then we can say more: ee is orthogonal to MM. For if

a h b e n c k d\array{a & \overset{h}{\to} & b\\ ^e\downarrow && \downarrow^n\\ c & \underset{k}{\to} & d}

is a commutative square with nMn\in M, then the pullback k *nk^*n is an MM-morphism through which ee factors. Hence k *nk^*n must be an isomorphism, and so the square admits a diagonal filler, which is unique since nMn\in M is monic. It follows that if all MM-images exist in CC, then MM is the right class of an orthogonal factorization system, and MM-images are precisely the factorizations in this OFS.

Conversely, it is easy to see that if (E,M)(E,M) is an OFS on a category CC, then all MM-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)(E,M) is a factorization system and ceamdc \overset{e}{\to} a \overset{m}{\to} d is an (E,M)(E,M)-factorization of f:cdf\colon c\to d, then not only is mm the MM-image of ff (the least MM-subobject through which ff factors), but dually ee is also the EE-coimage of ff, i.e. the greatest EE-quotient through which ff factors.

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

Construction using limits

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

Imflim(dd cd), Im f \simeq lim (d \rightrightarrows d \sqcup_c d) \,,

where d cdd \sqcup_c d denotes the pushout

c f d f d d cd. \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 RegMonoRegMono-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 codomain, while the regular coimage is the set-theoretic image topologized as a quotient space of the 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 MM-image for M=MonoM=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 1M_1 and M 2M_2 are two classes with M 1M 2M_1\subseteq M_2. If ff has both an M 1M_1-image im 1(f)im_1(f) and an M 2M_2-image im 2(f)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:

Lemma

If CC has finite limits and colimits, then there is a unique map

u:coimfimf u : coim f \to im f

from its regular coimage to its image such that

f=(ccoimfuimfd). f = (c \to coim f \stackrel{u}{\to} im f \to d) \,.
Proof

Because ff coequalizes c× dccc \times_d c \rightrightarrows c, a morphism hh in

c × dc c f d d cd epi h mono coimf imf \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 ccoimfc \to coim f is epi it follows that hh equalizes dd cdd \rightrightarrows d \sqcup_c d and hence uu in the diagram

c × dc c f d d cd epi h mono coimf u imf \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 uu is an isomorphism, then ff is sometimes called a strict morphism. In particular, if CC 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:

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 22-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 (,1)(\infty,1)-category theory

A (regular) (,1)(\infty,1)-image of a morphism f:cdf : c \to d in an (∞,1)-category with (∞,1)-limits and -colimits should be defined to be the (∞,1)-limit over the Cech co-nerve of ff:

imf:=lim (dd cdd cd cd). 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 in the case that the ambient (,1)(\infty,1)-category is just an ordinary category;

  • this implies that the inclusion imfdim f \to d is a regular monomorphism in the (,1)(\infty,1)-category sense (described here).

For more see n-image.

Examples

Applied to the (,1)(\infty,1)-category ∞Grpd this gives a notion of image of (∞,1)-functors between ∞-groupoids and hence a notion of image of functors between groupoids, 2-functors between 2-groupoids, etc.

Revised on February 18, 2014 01:20:18 by Anonymous Coward (92.40.250.76)