Category of Frames is Category
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
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
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$ Definition (a)