Total subobjects of quivers

Subobjects are categorically dual to quotients. In this context, if we were to dualize the idea of the lattice of thin congruences of a quiver, then its counterpart would have to be total subobjects. Let $Q$ be a quiver then a total quiver is one in which every object $o \in Ob(X)$ has at least one morphism going in to it $f : x \to o$ or going out of it $g : o \to y$. This produces the following duality:

• Subobjects: total subobjects can be determined from any morphism set
• Congruences: thin congruences can be determined from any output partition

The total function is an interior function on the lattice of subquivers $Sub(Q)$ whilst the thin function is a closure function on the lattice of congruences $Con(Q)$. We should always bear in mind that every concept in category theory has a categorical dual, so just as an object has congruences so too does it have subobjects.

See also:
Subobjects and quotients of thin quivers

External links:
Duality
Quivers