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A Segal condition is a (condition defining a) relation on a functor. In motivating cases these relations describe how a value of the functor may be constructed (up to equivalence) by values of subobjects- or truncated versions of .
A groupoid object in is a simplicial object in an (β,1)-category
that satisfies the groupoidal Segal conditions, meaning that for all and all partitions that share a single element , the (β,1)-functor exhibits an (β,1)-pullback
Write for the (β,1)-category of groupoid objects in , the full sub-(β,1)-category of simplicial objects on the groupoid objects.
An internal precategory in an -topos is a simplicial object in an (β,1)-category
such that it satifies the Segal condition, hence such that for all exhibits as the (β,1)-limit / iterated (β,1)-pullback
Write for the -category of internal pre-categories in , the full sub-(β,1)-category of the simplicial objects on the internal precategories.
An internal category in an -topos is an internal pre-category such that its core is in the image of the inclusion .
This is called a complete Segal space object in (Lurie, def. 1.2.10).
A directed graph is a presheaf
(β¦)
(description of this
diagram from
(β¦)
Complete Segal spaces were originally defined in
The relation to quasi-categories is discussed in
A survey of the definition and its relation to equivalent definitions is in section 4 of
See also pages 29 to 31 of
Jacob Lurie, On the Classification of Topological Field Theories
ncafΓ©, univalence is a Segal condition