Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
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.
The coimage of a morphism in a category is the image of the corresponding morphism in the opposite category .
In terms of colimits
If has finite limits and colimits, then the coimage of a morphism is the coequalizer of its kernel pair:
This is isomorphic to the pushout
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 -category
In an (∞,1)-category with (∞,1)-limits and -colimits, the colimit-definition of coimages generalizes as follows:
for a morphism in , let
be the Cech nerve of . This is the groupoid object in an (∞,1)-category that resolves the kernel pair equivalence relation: where is the relation that makes two generalized elements of equal if their image in 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 is the (∞,1)-colimit over this diagram
See also at infinity-image – As the ∞-colimit of the kernel ∞-groupoid.
- Morphisms for which image and coimage coincide (in a certain sense) are strict morphisms.
Let be a group. In the (∞,1)-category ∞-Grpd we have as a 0-truncated ∞-group object as well as its delooping , which is the one-object groupoid with as its morphisms.
Then: the coimage of the point inclusion is itself.
Because the homotopy-Cech nerve of the point inclusion is the usual simplicial incarnation of
now regarded as a simplicial object in ∞Grpd. Its homotopy colimit is again . 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.