So the notion can be understood as modelling the notion of an sSet-enrichment up to coherent homotopy, i.e. a weak enrichment. As such it is closely related to the notion of complete Segal space, which models the notion of an internal category in sSet.
Indeed, Segal categories may be considered with enrichment not just over sSet, but over other suitable model categories. In particular, an iterated enrichment over itself gives rise to the notion of Segal n-category which is a model for (∞,n)-categories.
Since the major difference between (small) -enriched categories and -internal categories is that in the first case the objects (as opposed to all the hom objects) form an ordinary set, while in the second these form an object of , too, accordingly a the definition of Segal category is like that of (complete) Segal space, only that the simplicial set of objects is required to be an ordinary set (a discrete simplicial set).
A Segal category is
such that is a discrete (= constant) simplicial set;
and such that the Segal maps
given by the above definition together with the remaining face map constitutes an ∞-anafunctor
given by the span
This encodes the composition operation in the Segal category .
The category of bisimplicial sets carries a model category structure whose fibrant objects are the Segal categories. This model structure for Segal categories is a presentation of the (∞,1)-category of (∞,1)-categories.
Then is a Segal category. Each simplicial set is discrete, for all , and all the morphisms
One may also form the -fold comma object-fiber product of a choice of base points with itself. This yields a Segal category incarnation of where in degree 1 we have the groupoid core of the arrow category of . For more on this see at Segal space – Examples - From a category.
The idea of Segal categories goes back (implicitly) to
They were named Segal categories in
An overview is on pages 164 to 169 of
A discussion with emphasis on the comparison of the various model category structures is in
The generalization to Segal n-categories is discussed in section 2 of
In the more general context of enriched (∞,1)-categories, this is discussed in
and in section 2 of