Category has Products and Equalizers iff Pullbacks and Terminal Object
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Theorem
Let $\mathbf C$ be a metacategory.
Then the following are equivalent:
- $(1): \mathbf C$ has all finite products and equalizers.
- $(2): \mathbf C$ has all pullbacks and a terminal object.
Proof
$(1)$ implies $(2)$
Suppose $\mathbf C$ has all finite products and equalizers.
That $\mathbf C$ has pullbacks follows from Pullback as Equalizer.
That $\mathbf C$ has a terminal object follows by Empty Product is Terminal Object.
$\Box$
$(2)$ implies $(1)$
Suppose $\mathbf C$ has all pullbacks and a terminal object.
From Category has Finite Products iff Terminal Object and Binary Products, it suffices to check that $\mathbf C$ has binary products.
That this is the case follows from Product as Pullback.
That $\mathbf C$ has equalizers follows from Equalizer as Pullback, as it is established that $\mathbf C$ has finite products.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 5.4$: Proposition $5.14$