Homotopy Type Theory conjunctive dagger 2-poset > history (Rev #2, changes)

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Contents

Idea

An essentially algebraic dagger 2-poset is a dagger 2-poset whose category of maps is a syntactic category for essentially algebraic theories? . “Cartesian” is a very overloaded term and has a different meaning for bicategories, and “finitely complete” dagger 2-posets understood in category-theoretic terms are simplysemiadditive dagger 2-posets.

Definition

An essentially algebraic dagger 2-poset is a dagger 2-poset CC such that

  • There is an object 0:Ob(C)0:Ob(C) such that for each object A:Ob(C)A:Ob(C), there is a functional dagger monomorphism i 0,A:Hom(0,A)i_{0,A}:Hom(0,A) such that for each object B:Ob(C)B:Ob(C) with a functional dagger monomorphism i B,A:Hom(B,A)i_{B,A}:Hom(B,A), there is a functional dagger monomorphism i 0,B:Hom(0,B)i_{0,B}:Hom(0,B).

  • For each object A:Ob(C)A:Ob(C), B:Ob(C)B:Ob(C), E:Ob(C)E:Ob(C) with functional dagger monomorphisms i B,A:Hom(B,A)i_{B,A}:Hom(B,A), i E,A:Hom(E,A)i_{E,A}:Hom(E,A), there is an object BE:Ob(C)B \cup E:Ob(C) with functional dagger monomorphisms i BE,A:Hom(BE,A)i_{B \cup E,A}:Hom(B \cup E,A), i B,BE:Hom(B,BE)i_{B,B \cup E}:Hom(B,B \cup E), i E,BE:Hom(E,BE)i_{E,B \cup E}:Hom(E,B \cup E), such that for every object D:Ob(C)D:Ob(C) with functional dagger monomorphisms i D,A:Hom(D,A)i_{D,A}:Hom(D,A) i B,D:Hom(B,D)i_{B,D}:Hom(B,D), i E,D:Hom(E,D)i_{E,D}:Hom(E,D), there is a functional dagger monomorphism i BE,D:Hom(BE,D)i_{B \cup E,D}:Hom(B \cup E,D).

Properties

Examples

The dagger 2-poset of sets and relations is an essentially algebraic dagger 2-poset.

See also

Revision on April 21, 2022 at 01:24:24 by Anonymous?. See the history of this page for a list of all contributions to it.