nLab globular set

Globular sets

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higher category theory

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Extra properties and structure

1-categorical presentations

Globular sets

Idea

Globular sets are to simplicial sets as globes are to simplices.

They are one of the major geometric shapes for higher structures: if they satisfy a globular Segal condition then they are equivalent to strict ∞-categories.

Definition

Basic definition

Definition

The globe category 𝔾\mathbb{G} is the category whose objects are the natural numbers, denoted here [n][n] \in \mathbb{N} (N.B. not to be confused with ordinals in any structural sense) and whose morphisms are generated from

σ n:[n][n+1] \sigma_n : [n] \to [n+1]
τ n:[n][n+1] \tau_n : [n] \to [n+1]

for all nn \in \mathbb{N} subject to the relations (dropping obvious subscripts)

σσ=τσ \sigma\circ \sigma = \tau \circ \sigma
στ=ττ. \sigma\circ \tau = \tau \circ \tau \,.
Definition

A globular set, also called an ω\omega-graph, is a presheaf on 𝔾\mathbb{G}. The category of globular sets is the category of presheaves

gSetPSh(𝔾). gSet \coloneqq PSh(\mathbb{G}) \,.
Remark

This means that a globular set XgSetX \in gSet is given by a collection of sets {X n} n\{X_n\}_{n \in \mathbb{N}} (the set of nn-globes) equipped with functions

{s n,t n:X n+1X n} n \{s_n,t_n \colon X_{n+1} \to X_n\}_{n \in \mathbb{N}}

called the nn-source and nn-target maps (or similar), such that the globular identities hold: for all nn \in \mathbb{N}

  • s ns n+1=s nt n+1s_n \circ s_{n+1} = s_n \circ t_{n+1}

  • t ns n+1=t nt n+1t_n \circ s_{n+1} = t_n \circ t_{n+1} .

Remark

The globular identities ensure that two sequences of boundary maps

f nf n+m1f n+m:S n+m+1S n f_n \circ \cdots \circ f_{n+m-1} \circ f_{n+m} : S_{n+m+1} \to S_n

with n,mn,m \in \mathbb{N} and for f k,{s k,t k}f_k, \in \{s_k, t_k\} are equal if and only if their last term f nf_n coincides; for all n,mn,m \in \mathbb{N} we have

s ns n+1s n+mi n+mi n+1i n=Id s_n \cdots s_{n+1} \circ \cdots \circ s_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n = \mathrm{Id}
t nt n+1t n+mi n+mi n+1i n=Id. t_n \cdots t_{n+1} \circ \cdots \circ t_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n = \mathrm{Id} \,.

For SS a globular set we may therefore write unambiguously

s n,t n:S n+m+1S n s_n, t_n : S_{n+m+1} \to S_n
i n:S nS n+m+1 i_n : S_n \to S_{n+m+1}

with i n,s n,t mi_n, s_n, t_m the sequence of mm consecutive identity-assigning, source or target maps, respectively.

Remark

The presheaf definition can understood from the point of view of space and quantity: a globular set is a space characterized by the fact that and how it may be probed by mapping standard globes into it: the set S nS_n assigned by a globular set to the standard nn-globe [n][n] is the set of nn-globes in this space, hence the way of mapping a standard nn-globe into this spaces.

More generally:

Definition

A globular object XX in a category 𝒞\mathcal{C} is a functor X:𝔾 op𝒞X : \mathbb{G}^{\mathrm{op}} \to \mathcal{C}.

Reflexive globular sets

If to the globe category we add additional generating morphisms

ι n:[n+1][n] \iota_n : [n+1] \to [n]

satisfying the relations

ισ=Id \iota \circ \sigma = \mathrm{Id}
ιτ=Id \iota \circ \tau = \mathrm{Id}

we obtain the reflexive globe category, a presheaf on which is a reflexive globular set. In this case the morphism

i n:=S(ι n):S nS n+1 i_n := S(\iota_n) : S_{n} \to S_{n+1}

is called the nnth identity assigning map; it satisfies the globular identities:

si=Id s \circ i = \mathrm{Id}
ti=Id t \circ i = \mathrm{Id}

nn-globular sets

A presheaf on the full subcategory of the globe category containing only the integers [0][0] through [n][n] is called an nn-globular set or an nn-graph. An nn-globular set may be identified with an \infty-globular set which is empty above dimension nn.

Note that a 11-globular set is just a directed graph, and a 00-globular set is just a set.

Examples

  • Any strict 2-category or bicategory has an underlying 2-globular set. Likewise, any tricategory has an underlying 3-globular set. Globular sets can be used as underlying data for n-categories as well; see for instance Batanin ∞-category.

  • A strict omega-category is a globular set CC equipped in each degree with the structure of a category such that for every pair k 1<k 2k_1 \lt k_2 \in \mathbb{N} the induced structure on the 2-graph C k 2C k 1C 0C_{k_2} \stackrel{\to}{\to} C_{k_1} \stackrel{\to}{\to} C_0 is that of a strict 2-category.

  • The globular nn-globe G nG_n is the globular set represented by nn, i.e. G n():=Hom G(,n)G_n(-) := Hom_G(-,n).

Grothendieck homotopy theory

The category of globes is not a weak test category according to Scholium 8.4.14 in Cisinski 06.

However, if we construct the free strict monoidal category on the category of globes, while ensuring that the terminal object becomes the monoidal unit, then the resulting category of polyglobes is a test category.

References

The definition is reviewed around def. 1.4.5, p. 49 of

See also

The definition of globular set, without using that term, is in 2.2 and 2.3 of the following paper:

  • Ronnie Brown and Philip J. Higgins, “The equivalence of \infty-groupoids

    and crossed complexes“, Cah. Top. G'eom. Diff. 22 (1981) 371-386.

The following paper constructs from the cubical case a strict globular ω\omega-groupoid of a filtered space:

  • Ronnie Brown“A new higher homotopy groupoid: the fundamental globular

    ω\omega-groupoid of a filtered space“, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.

Last revised on October 12, 2022 at 08:17:24. See the history of this page for a list of all contributions to it.