The Segal condition is a condition on a simplicial object which says that each component is obtained from by the -fold fiber product of over which “glues” copies of end-to-end.
Hence if one thinks of as a collection of objects and of as a collection of morphisms, then satisfies the Segal condition precisely if each can be interpreted as the collection of sequences of composable morphisms of length . The precise formulation is below in Definition – For simplicial objects.
Accordingly, if Set is the category of sets, then the Segal condition characterizes precisely those simplicial sets which are the nerve of a small category, theorem 1 below. This is the observation due to (Segal 1968), following Grothendieck, which today gives the Segal condition its name. Sometimes this statement also called the nerve theorem (no relation to what is called nerve theorem in homotopy theory).
It is useful to decompose this statement into its constituents as follows:
A small category may be thought of as a directed graph equipped with a unital associative composition operation. This corresponds to a sequence of inclusions of sites
into the simplex category, where is the category of finite non-empty directed linear graphs:
a directed graph is equivalently a presheaf on ;
a presheaf on , hence a simplicial set encodes via its face and degeneracy maps a kind of associative and unital composition – but not necessarily “of composable morphisms” if is not given in the above fashion.
In terms of this we can say that equipping a directed graph with the structure of a category is equivalent to asking for its pushforward along (which encodes all the collections of sequences of composable edges) to be equipped with a lift to a simplicial object through the pullback along . Conversely, the simplicial objects obtained as such lifts are precisely the simplicial objects that satisfy the Segal condition.
Formulated in this way one sees that the Segal condition has a large variety of generalizations to structures with richer kinds of composition operations, such as globular operads. This is made precise below in Definition – For cellular objects.
For simplicial objects
Let be a category with pullbacks.
A simplicial object
is said to satisfy the Segal condition if it sends the colimits in the simplex category to limits, hence if the Segal maps exhibit equivalences
for all .
More in detail:
For all , consider naturally as a cocone in the simplex category under the diagram
with copies of at the bottom, such that the cocone injection of the th copy is .
A simplicial object satisfies the Segal conditions if it sends these cocones to limit cones in .
For cellular objects
A globular theory is a wide subcategory inclusion
of the globular site . There is an equivalence of categories
of ω-graphs and sheaves on the globular site. In particular for the cell category (Theta category) a presheaf on is a cellular object.
The Segal condition on a cellular object is that the restriction to the cellular site is a sheaf there.
The cellular objects that satisfy the Segal condition are precisely the ω-category objects (Berger).
The cellular spaces/ cellular simplicial sets/cellular ∞-groupoids that satisfy the Segal condition as a weak homotopy equivalence/ equivalence of ∞-groupoids is a Theta_n-space? an (∞,n)-category.
Characterization of nerves of (higher) categories
Of simplicial nerves of small categories
The archetypical role of the Segal condition is to make the following statement true.
This is due to (Segal 1968), following Grothendieck.
Of complete Segal spaces
By refining the above result from sets to -groupoids, one obtains the pre-category object in an (infinity,1)-category.
Of cellular nerves of strict -categories
Similarly, a cellular set is the cellular nerve of a strict omega-category precisely if it satisfies the cellular Segal condition. (Berger).
Of cellular models of -categories
See at Theta-space.
In terms of sheaf conditions
We discuss an equivalent formulation of the Segal condition in terms of notions in topos theory/(∞,1)-topos theory. This perspective for instance lends itself more to a formulation of Segal conditions in terms of the internal language of toposes.
For simplicial objects and category objects
We characterize below in prop. 6 the category of categories as the pullback of the topos of simplicial set along the inclusion of the topos of graphs into that of presheaves on finite linear graphs.
First we state some preliminaries.
Therefore consider instead the following:
be the full subcategory of that of directed graphs on the linear graphs for .
for the full subcategory on the linear graphs with no edge and with one edge.
for the adjoint triple induced on categories of presheaves by the inclusion of def. 4: is given by precomposition with , is left and is right Kan extension along .
The functor of def. 2 sends a graph to the presheaf which on is given by th iterated pullback
and which sends an inclusion to the corresponding projection map out of the pullback.
We may call the nerve of the graph .
Using the Yoneda lemma and the defining hom-isomorphisms of the adjunction as well as the fact that the hom functor sends colimits in the first argument to limits, we have
For declare a unique covering family of to be
Then this is a coverage on .
A presheaf is a sheaf with respect to the coverage of def. 4 precisely if it is in the essential image of the graph-nerve functor
of prop. 3. This yields an equivalence of categories
with the category of sheaves on . The graph-nerve functor is a full and faithful functor
for the adjoint triple induced on categories of presheaves by the inclusion of def. 3: is given by precomposition with , is left and is right Kan extension along .
In terms of all this the nerve theorem 1 says the following:
We have geometric morphisms of toposes
which capture the Segal condition as follows.
The commuting diagram of 1-categories
is a pullback.
For cellular objects and -category objects
The immediate generalization of prop. 6 from simplicial objects to cellular objects is the following.
be the defining inclusion of the cellular site into the cell category.
The category of strict ω-categories is the pullback
See at globular theory for more.
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 they are attributed to Alexander Grothendieck.
The interpretation of the Segal condition as a sheaf condition is reviewed for instance in section 2 of
and discussed for strict infinity-categories in
- Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)
Based on that, an iterative and homotopy-theoretic formulation of the cellular Segal conditions is in section 5 of