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globular set

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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 ω-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 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 ω-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).

References

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

See also

  • Sjoerd Crans, On combinatorial models for higher dimensional homotopies (web)

Revised on January 16, 2014 04:36:01 by Todd Trimble (67.81.95.215)