Category of Frames is Category

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Theorem

Let $\mathbf{Frm}$ denote the category of frames.


Then:

$\mathbf{Frm}$ is a metacategory

Proof

Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.


For any two frame homomorphisms their composition (in the usual set theoretic sense) is again a frame homomorphism by Composite Frame Homomorphism is Frame Homomorphism.


For any frame $L = \struct{S, \preceq}$, we have the identity mapping $\operatorname{id}_S$.

From Identity Mapping is Frame Homomorphism we have $\operatorname{id}_S$ is a frame homomorphism.

By Identity Mapping is Left Identity and Identity Mapping is Right Identity, this is the identity morphism for $L$.


Finally by Composition of Mappings is Associative, the associative property is satisfied.


Hence $\mathbf{Frm}$ is a metacategory.

$\blacksquare$


Sources

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