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Saturday, October 8, 2022

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

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