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internal formulation of cohesionAn adjoint quadruple
The exhibits adjoint triple induced by the original adjoint quadruple (where in fact is not used in the basic definition, only in some applications) defining infinitesimal cohesion is denoted by
cohesion, if and are full, faithful and preserves finite products
infinitesimal cohesion, if and are full, faithful and preserves the terminal object. (In some cases we do not care about .)
The adjoint triples induced by these adjoint quadruples we denote by
Let , be cohesive toposes. An adjoint triple
in the cohesive case by
in the infinitesimal cohesive case by
is The said encoding to of cohesion in Ho TT proceeds in the following stepsexhibit an infinitesimal cohesion if is full, faithful and preserves the terminal object.
(1) It follows from the definition that also is full and faithful.
(2) is a (geometric) morphism of toposes over . This means that
commutes.
(3) The same holds for
(1a) (resp. ) determines a reflective subcategory. In “independent” type theory, the axioms for this are displayed in internal formulation of cohesion. However if we consider these axioms in Ho TT (which is a dependent type theory) these axioms do not define a reflective subcategory but a reflective subfibration. This means that we need a translation of the notion of reflective subfibration back to that of reflective subcategory. This translation is:
(1b)
We proceed by encoding the following partial notions:
(resp. ) determines a coreflective subcategory.
preserves the terminal object.
(resp. ) determines a reflective subcategory
(resp. ) determines a coreflective subcategory. (In some cases we do not care about this last axiom.)
(1) It follows from the definition that also is full and faithful.
(2) is a (geometric) morphism of toposes over . This means that
commutes.
(3) The same holds for