nLab n-fold complete Segal space

Idea

$n$-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. $n$-fold Segal spaces similarly result from viewing $(\infty,n)$-categories as a special class of $n$-fold internal $(\infty,1)$-categories in ∞-groupoids.

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

To describe the correct homotopy theory of $(\infty,n)$-categories we also want to regard the fully faithful and essentially surjective morphisms between $n$-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.

Definition

$n$-fold Segal objects

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

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

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

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

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

1. The $(n-1)$-simplicial object $X_0 : \Delta^{n-1,op} \to \mathcal{C}$ is constant
2. The $(n-1)$-simplicial objects $X_i$ are $(n-1)$-fold Segal objects for all $i$.

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

Complete $n$-fold Segal spaces

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

Definition

A morphism $X \to Y$ between $n$-fold Segal spaces is fully faithful and essentially surjective if:

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

Definition

An $n$-fold Segal space $X$ is complete if:

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

Remark

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

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

Theorem

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

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

Definition via the model category of simplicial sets

(…)

Related concepts

• Theta-spaces, Segal n-categories

• higher Segal space

• semi-Segal space

• weakly globular n-fold category

References

The definition originates in the thesis

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

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

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (arXiv:1112.0040, slides)

The basic idea was being popularized and put to use in

• Jacob Lurie, On the Classification of Topological Field Theories

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

• Jacob Lurie, $(\infty,2)$-Categories and the Goodwillie Calculus I (arXiv:0905.0462)

A Quillen adjunction relating $n$-complicial sets to $n$-fold complete Segal spaces:

• Viktoriya Ozornova, Martina Rovelli, A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories, Advances in Mathematics 421 (2023) 108980 [doi:10.1016/j.aim.2023.108980]

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