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Idea
Mike Shulman’s dependently typed first-order theory SEAR.
Definition
A model of SEAR consists of
A type , whose terms are called objects. “sets”.
For each object “set”, a set , whose terms are called elements. “elements”.
For each object “set” and , a set , whose elements terms are called morphisms. “relations”.
For every each object “set” and , there is a function
For every each object “set”, , , there is a function
such and that a term
Let where the type of all maps in be is defined the as propositional truncation of the type.
For each object and , there is a function such that is contractible.
There exists an object with a term , where is the propositional truncation of the type .
For every object and and morphism , there is an object and maps , , such that and for two elements and , and imply .
For every object , there is an object and a morphism such that for each morphism , there exists a map such that .
There is an object with maps and , such that for every object with maps and , there is a map such that and for all elements , .
For every object and function , there exists an object with a map and a -indexed family of objects such that (1) for every element , is contractible and (2) for every element , if there exists an object such that is contractible, then is in the image of .
Let the type of all functional entire “relations” between “sets” and in be defined as
For each “set” and , a function and a term .
A “set” with a term .
For each “set” and and “relation” , a “set” and functional entire “relation” , , and a term and a term
For each “set” , a “set” and a “relation” and a term
A “set” with functional entire “relations” and and a term
For each “set” and function , a “set” with a functional entire “relation” , a function and terms