I was filling out a description box for a potential meeting with a contact, & I went to type the sentence:
"To have a follow up conversation about ____'s guest lecture."
Would that be considered a full sentence or a fragment?
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Yes, it's a sentence fragment because it's missing a subject, verb, or both. Yes, "have" is a verb, but yes, it's a "to-infinitive" and is not the main verb of the sentence, so its conjugation doesn't have a bearing on the question.
Is that a problem? Not necessarily. The context is informal and the meaning is understood. The implied sentence is something like:
[The reason for the meeting is] to have a follow-up conversation...