A Segal space is a pre-category object in ∞Grpd.
A genuine category object in ∞Grpd is a complete Segal space. This is a way of speaking of (∞,1)-categories.
A Segal space is a simplicial topological space or bisimplicial set which satisfies the Segal conditions:
for all the square
is a homotopy pullback square.
A Segal space for which is a discrete space is called a Segal category. See there for more dicussion.
For a (small) category we may regard its ordinary nerve simplicial set as a Segal space, under the canonical inclusion ,
In fact, the classical “nerve theorem” about the Segal conditions says that a simplicial set is the nerve of a category precisely if it is a Segal space.
Notice that is precisely the subset of isomorphisms in all morphisms of .
Therefore under this identification, is a complete Segal space precisely if is a gaunt category, hence precisely if the only isomorphisms in are the identities.
In particular if is a (0,1)-category, hence a preordered set, then is complete Segal precisely if is in fact an partially ordered set.
Construction in from a category
Let be an ordinary category. We discuss how Segal spaces are associated with this.
Let be a groupoid and a functor which is essentially surjective.
Then let be the “lax fiber product” of with itself, or rather the comma object of with itself, hence the comma category, sitting in the universal square
Next let be the “3-fold comma category”, hence the comma category in
and so forth: .
This way an object of is an -tuple of objects together with a sequence of composable morphisms , and a morphism is an -tuple of morphisms and a pasting commuting diagram
By direct inspection, the maps obtained this way are isofibrations, hence fibrations in the canonical model structure on Grpd and so the homotopy pullbacks that enter the Segal conditions for are given by ordinary fiber products. These clearly satisfy the Segal conditions, hence
constructed this way is a Segal space.
Two special case of the functor are important:
if is the core of and is the canonical core inclusion, one finds that by the above construction is , the arrow category of the core of . This is equivalent to by, for instance, the source or restriction map. Hence for the core inclusion, the above construction gives the complete Segal space corresponding to the category .
if is a choice of basepoints in each isomorphism class of , then is the Segal category incarnation of the category .
We consider the situation of From a category, but now conversely, starting with a Segal space in groupoids and then extracting a category from it.
Consider a Segal space that is degreewise just a 1-groupoid, hence a simplicial object in the inclusion
Choosing this to be Reedy fibrant, the map is an isofibration.
We may write an object as a horizontal morphism
and a morphism in as a vertical double category arrow:
Then the fact that is an isofibration means that for every “niche”
namely for every pair of morphisms in and lift of its codomain to an object , there is a “niche filler”
namely a lift of the whole pair to a morphism in , and this is necessarily universal in that any other such lift uniquely factors through this one (because is a groupoid).
Comparison with the definition of a 2-category equipped with proarrows in the incarnation as a double category shows that this is the beginning of the construction of a pseudo double category whose vertical category is and whose weak horizontal composition is that induced by the Segal maps.
Assume next that is a 1-monomorphism, as are all the higher , for , hence that is 2-coskeletal as a simplicial object. This means that the horizontal composition in this pseudo double category has unique composites, hence that the horizontal category is an ordinary category. If then furthermore the composite is an equivalence, hence is the Segal space is a complete Segal space this means that arises from this horizontal category by the construction above.
See generally the references at complete Segal space.
The “Segal conditions” are first discussed in
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 105–112 (1968),
where it is attributed to Alexander Grothendieck.
The term “Segal space” is due to
The invertible case of Segal spaces, hence models for groupoid objects in an (infinity,1)-category are discussed in section 3 of
- Julia Bergner, Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)