n-fold complete Segal space


Higher category theory

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Internal (,1)(\infty,1)-Categories



nn-fold complete Segal spaces are a model for (∞,n)-categories, i.e. the homotopical version of n-categories.

We can view strict n-categories as n-fold categories where part of the structure is trivial; for example, strict 2-categories can be described as double categories where the only vertical morphisms are identities. nn-fold Segal spaces similarly result from viewing (,n)(\infty,n)-categories as a special class of nn-fold internal (,1)(\infty,1)-categories in ∞-groupoids.

If 𝒞\mathcal{C} is an (,1)(\infty,1)-category, then (,1)(\infty,1)-categories internal to 𝒞\mathcal{C} can be defined as certain simplicial objects in 𝒞\mathcal{C} (namely those satisfying the “Segal condition”). Thus nn-fold internal (,1)(\infty,1)-categories in \infty-groupoids correspond to a class of nn-simplicial \infty-groupoids, and nn-fold Segal spaces are defined by additionally specifying certain constancy conditions.

To describe the correct homotopy theory of (,n)(\infty,n)-categories we also want to regard the fully faithful and essentially surjective morphisms between nn-fold Segal spaces as equivalences. It turns out that, just as in the case of Segal spaces, the localization at these maps can be accomplished by restricting to a full subcategory of complete objects.


nn-fold Segal objects

If 𝒞\mathcal{C} is an (,1)(\infty,1)-category with pullbacks, we say that a simplicial object X :Δ op𝒞X_\bullet : \Delta^{op} \to \mathcal{C} satisfies the Segal condition if the squares

X m+n X n X m X 0 \array{ X_{m+n} &\to& X_{n} \\ \downarrow && \downarrow \\ X_{m} &\to& X_{0} }

are all pullbacks. Such Segal objects give the (,1)(\infty,1)-categorical version of internal categories as algebraic structures. (I.e. we have not inverted a class of fully faithful and essentially surjective morphisms.)

If Seg(𝒞)Seg(\mathcal{C}) denotes the full subcategory of Fun(Δ op,𝒞)Fun(\Delta^{op}, \mathcal{C}) spanned by the Segal objects, then this is again an (,1)(\infty,1)-category with pullbacks, so we can iterated the definition to obtain a full subcategory Seg n(𝒞)Seg^{n}(\mathcal{C}) of Fun(Δ n,op,𝒞)Fun(\Delta^{n,op}, \mathcal{C}) of Segal Δ n\Delta^{n}-objects in 𝒞\mathcal{C}.

We can now inductively define nn-fold Segal objects by imposing constancy conditions: An nn-fold Segal object in 𝒞\mathcal{C} is a Segal Δ n\Delta^{n}-object XX such that

  1. The (n1)(n-1)-simplicial object X 0:Δ n1,op𝒞X_0 : \Delta^{n-1,op} \to \mathcal{C} is constant
  2. The (n1)(n-1)-simplicial objects X iX_i are (n1)(n-1)-fold Segal objects for all ii.

When 𝒞\mathcal{C} is the (,1)(\infty,1)-category of spaces (or \infty-groupoids) we refer to nn-fold Segal objects as nn-fold Segal spaces.

Complete nn-fold Segal spaces

We now define fully faithful and essentially surjective morphisms between nn-fold Segal inductively in terms of the corresponding notions for Segal spaces:


A morphism XYX \to Y between nn-fold Segal spaces is fully faithful and essentially surjective if:

  1. X ,0,,0Y ,0,,0X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0} is a fully faithful and essentially surjective morphism of Segal spaces
  2. X 1,,,Y 1,,,X_{1,\bullet,\ldots,\bullet} \to Y_{1,\bullet,\ldots,\bullet} is a fully faithful and essentially surjective morphism of (n1)(n-1)-fold Segal spaces

An nn-fold Segal space XX is complete if:

  1. The Segal space X ,0,,0X_{\bullet,0,\ldots,0} is complete.
  2. The (n1)(n-1)-fold Segal space X 1,,,X_{1,\bullet,\ldots,\bullet} is complete.

There are several equivalent ways to reformulate these inductive definitions. For example, a morphism f:XYf : X \to Y is fully faithful and essentially surjective if and only if:

  1. X ,0,,0Y ,0,,0X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0} is essentially surjective.
  2. For all objects x,xX 0,,0x,x' \in X_{0,\ldots,0} the induced map on (n1)(n-1)-fold Segal spaces of morphisms X(x,x)Y(fx,fx)X(x,x') \to Y(fx,fx') is fully faithful and essentially surjective.

The complete nn-fold Segal spaces are precisely the nn-fold Segal spaces that are local with respect to the fully faithful and essentially surjective morphisms. Thus the localization of the (,1)(\infty,1)-category of nn-fold Segal spaces at this class of morphisms is equivalent to the full subcategory of complete nn-fold Segal spaces.

This was first proved in Barwick’s thesis, generalizing Rezk’s proof in the case n=1n=1. Later, Lurie extended the notion of complete Segal objects to more general contexts than spaces, which allows an inductive definition of complete nn-fold Segal spaces as complete Segal objects in complete (n1)(n-1)-fold Segal spaces. The theorem for nn-fold Segal spaces then follows by inductively applying the generalization of Rezk’s theorem (for the case n=1n=1) to this setting.

Definition via the model category of simplicial sets



The definition originates in the thesis

  • Clark Barwick, (,n)(\infty,n)-CatCat as a closed model category PhD (2005)

which however remains unpublished. It appears in print in section 12 of

The basic idea was being popularized and put to use in

A detailed discussion in the general context of internal categories in an (∞,1)-category is in section 1 of

For related references see at (∞,n)-category .

Last revised on March 2, 2018 at 15:52:37. See the history of this page for a list of all contributions to it.