coimage
The coimage of a morphism is the notion dual to its image.
Under certain conditions (the codomain of) the coimage coincides with (the domain of) the image, and even if not, often the coimage is what one calls (or wants to think of as) the image. The two notions tend to go hand in hand, with the coimage being an epi and the image being a mono in some (epi, mono) factorization. For more of the general theory see image.
The coimage of a morphism $f : c \to d$ in a category $C$ is the image of the corresponding morphism in the opposite category $C^{op}$.
If $C$ has finite limits and colimits, then the coimage of a morphism $f : c \to d$ is the coequalizer of its kernel pair:
This is isomorphic to the pushout $c \sqcup_{c\times_d c} c$
So in
the outer square is a pullback square while the inner is a pushout.
Notice that being a coequalizer, the morphism
is an epimorphism and in fact a regular epimorphism.
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
be the Cech nerve of $f$. This is the groupoid object in an (∞,1)-category that resolves the kernel pair equivalence relation: where $c \times_d \stackrel{\to}{\to} c$ 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
The coimage of $f$ is the (∞,1)-colimit over this diagram
See also at infinity-image – As the ∞-colimit of the kernel ∞-groupoid.
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 $\mathbf{B}G$, which is the one-object groupoid with $G$ as its morphisms.
Then: the coimage of the point inclusion $f : * \to \mathbf{B}G$ is $\mathbf{B}G$ itself.
Because the homotopy-Cech nerve of the point inclusion is the usual simplicial incarnation of $G$
now regarded as a simplicial object in ∞Grpd. Its homotopy colimit is again $\mathbf{B}G$. 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.