Fibre product is the equalizer of a product
Suppose products and equalizers exist in
Then the fibre product
where
Proof
Let
π΄ π§ 1 β΅ π π§ 2 βΆ π΅ such that
, so π π§ 1 = π π§ 2 where ( π§ 1 , π§ 2 ) : π β π΄ Γ π΅ π π 1 ( π§ 1 , π§ 2 ) = π π 2 ( π§ 1 , π§ 2 ) . Now there exists
so that π’ : π β πΈ . Thus e q π’ = ( π§ 1 , π§ 2 ) π 1 π’ = π 1 e q π’ = π 1 ( π§ 1 , π§ 2 ) = π§ 1 , π 2 π’ = π 2 e q π’ = π 2 ( π§ 1 , π§ 2 ) = π§ 2 . Given an alternate
with the property π’ β² : π β πΈ , then π π π’ β² = π§ π so π π e q π’ β² = π§ π , and since the equalizer is monic e q π’ β² = ( π§ 1 , π§ 2 ) = e q π’ . π’ = π’ β²
Footnotes
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2010. Category theory, ΒΆ5.5, pp. 93β94 β©