Equalizer

up:: 027 MOC Category Theory

Fonte: Equaliser (mathematics) - Wikipedia

Given objects in some category , and morphisms , then their equalizer is

1. an object , and
2. a morphism

through which any other morphism , for any other object factors through , i.e.

The idea is that "" is made possible through this categorical object. to-be-elaborated

Its dual is a coequalizer.

Examples

Linear map’s kernel in vector spaces

Given a Linear Operator between vector spaces, then its kernel can be thought to be an equalizer between and the map () — or rather, an equalizer’s object, whose morphism would be . We have the diagram

Let some other such that it also has some linear map that satisfies the above, i.e.

Then, it’ll be true that¹

Since that’s the case, we can simply use the very same map to connect to :

Sheaves over Set

Given a Presheaf over a Topological Space onto Set, it’ll be a Sheaf over when, for any open sets and open covers of these given ‘s, the following equalizer diagram commutes:

Given and an open cover of it, , let be an element of (the set) . to-be-elaborated

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References

• III EBCT (Encontro Brasileiro em Teoria das Categorias) - 25/05 — ~01:47:00
• equalizer in nLab
• quiver: a modern commutative diagram editor

Footnotes

1. Remember that . ↩