Complete Segal spaces are one model for (∞,1)-categories.
The rough idea is that a complete Segal space is the nerve of a category enriched weakly over Top/sSet: it is not a simplicial set, but a simplicial topological space or bisimplicial set that satisfies the homotopy theoretic analog of the condition that otherwise implies that a simplicial set is the nerve of a category.
A bit more precisely: to determine if a simplicial set arises from a category by passing to its nerve one has to check the Segal conditions: that for all natural numbers the square
is a pullback square in Set (where the maps project out the indicated parts of the objects of the simplex category regarded as posets). is the nerve of a category precisely if this is the case for all .
This condition is internalized homotopically in the category of spaces to get the definition of a Segal space. One can interpret “spaces” here as meaning either (sufficiently nice) topological spaces or simplicial sets; in the latter case a Segal space is a particular sort of bisimplicial set.
A Segal space is a simplicial nice topological space/simplicial set which is Reedy fibrant and for which for all the square
is a homotopy pullback square in Top/sSet.
Next, the idea is that, roughly, a Segal space is a complete Segal space if the the fundamental ∞-groupoid of is the maximal topological groupoid contained in the topologically enriched category of which is like the nerve of.
More precisely…
…
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