## Internal Parameterization of Hyperconnected Quotients

### Ryuya Hora – 19 April 2023

This talk focuses on the speaker's preprint, "Internal Parameterization of Hyperconnected Quotients." The main theorem establishes a bijective correspondence between hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos) and certain "internal" structures.

This talk focuses on the speaker's preprint, "Internal Parameterization of Hyperconnected Quotients." The main theorem establishes a bijective correspondence between hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos) and certain "internal" structures.

This talk is divided into three parts:

1: The first problem in Lawvere's Open Problems: We will briefly review the fundamentals of topos theory and introduce Lawvere's open problem related to what he calls "Quotient toposes."

2: Hyperconnected Quotients: We present a partial solution to the open problem, specifically addressing the subclass known as hyperconnected quotients.

3: Local State Classifiers: Finally, we introduce the central tool for our theorem, a local state classifier, which is simply defined as "the colimit of all monomorphisms." Through examples, we will clarify why such an object works and the intuition behind it.

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