The quality or state of being an archbishop, or the authority and jurisdiction of an archbishop.
From 'archiepiscopal' plus the suffix '-ity' meaning 'condition or quality,' creating an abstract noun to describe the position or its inherent characteristics.
English has this amazing ability to turn adjectives into abstract nouns with '-ity'—'archiepiscopality' sounds fancy but it just means 'being an archbishop' or 'archbishop-ness,' which is how academic and religious language gets so wonderfully unwieldy.
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