nLab
coimage

• coimage

• image

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Idea

The coimage of a morphism is the notion dual to its image.

Under certain conditions coimages coincide with images and even if not, often the coimage is what one wants to think of as the image. You cannot have an image or coimage without the other. For more of the general theory see image.

Definition

The coimage of a morphism $f:c\to d$ in a category $C$ is the image of the corresponding morphism in the opposite category $C^{\mathrm{op}}$.

In terms of colimits

If $C$ has finite limits and colimits, then the coimage of a morphism $f:c\to d$ is the coequalizer of its kernel pair:

$$\mathrm{coim}f\simeq \mathrm{colim}(c{\times }_{d}c\overrightarrow{\to }c),$$coim f \simeq colim ( c \times_d c \stackrel{\to}{\to} c) \,,

This is isomorphic to the pushout $c{\bigsqcup }_{c{\times }_{d}c}c$

$$\mathrm{coim}f\simeq c{\bigsqcup }_{c{\times }_{d}c}c.$$coim f \simeq c \sqcup_{c\times_d c} c \,.

So in

$$\begin{array}{ccccc}c{\times }_{d}c& & \to & & c\\ & & & \swarrow \\ {\downarrow }^f& & \mathrm{coim}f& & {\downarrow }^f\\ & \nearrow & & \searrow \\ C& & \stackrel{f}{\to }& & d\end{array}$$\array{ c \times_d c &&\to&& c \\ &&& \swarrow \\ \downarrow^f && coim f && \downarrow^f \\ & \nearrow && \searrow \\ C && \stackrel{f}{\to} && d }

the outer square is a pullback square while the inner is a pushout.

Notice that being a coequalizer, the morphism

$$c\to \mathrm{coim}f$$c \to coim f

is an epimorphism and in fact a regular epimorphism.

In an $(\mathrm{\infty },1)$-category

In an (∞,1)-category $C$ with (∞,1)-limits and -colimits, the colimit-definition of coimages generalizes as follows:

for $f:c\to d$ a morphism in $C$, let

$$C(f)=\left(\cdots \overrightarrow{\overrightarrow{\to }}c{\times }_{d}c\overrightarrow{\to }c\right)$$C(f) = \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} c \times_d c \stackrel{\to}{\to} c \right)

be the Cech nerve of $f$. This is the groupoid object in an (∞,1)-category that resolves the kernel pair equivalence relation: where $c{\times }_d\overrightarrow{\to }c$ is the relation that makes two generalized elements of $c$ equal if their image in $d$ is equal, the full Cech nerve is the internal ∞-groupoid where there is just an equivalence between such two elements.

The Cech nerve is a simplicial diagram

$$C(f):{\Delta }^{\mathrm{op}}\to C$$C(f) : \Delta^{op} \to C

The coimage of $f$ is the (∞,1)-colimit over this diagram

$$\mathrm{coim}(f):=\underrightarrow{\lim }\left(\cdots \overrightarrow{\overrightarrow{\to }}c{\times }_{d}c\overrightarrow{\to }c\right)$$coim(f) := \lim_\to \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} c \times_d c \stackrel{\to}{\to} c \right)

See also at infinity-image – As the ∞-colimit of the kernel ∞-groupoid.

Properties

• Morphisms for which image and coimage coincide (in a certain sense) are strict morphisms.

Examples

For $(\mathrm{\infty },1)$-coimages

Let $G$ be a group. In the (∞,1)-category $C=$ ∞-Grpd we have $G$ as a 0-truncated ∞-group object as well as its delooping $BG$, which is the one-object groupoid with $G$ as its morphisms.

Then: the coimage of the point inclusion $f:*\to BG$ is $BG$ itself.

Because the homotopy-Cech nerve of the point inclusion is the usual simplicial incarnation of $G$

$$C(f)=(\cdots G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$C(f) = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right)

now regarded as a simplicial object in ∞Grpd. Its homotopy colimit is again $BG$. This follows for instance abstractly from the fact that ∞Grpd is an (∞,1)-topos and therefore satisfies Giraud's axioms, which say that every groupoid object in an (∞,1)-category is effective in an (∞,1)-topos.