globular set (changes) in nLab

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Globular sets

Idea

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

Much like simplicial sets underly common geometric definitions of higher categories, so globular sets underly some algebraic definitions of higher categories, see below.

Definition

Basic definition

More generally:

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

ι∘σ=Id \iota \circ \sigma = \mathrm{Id}

ι∘τ=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 n→S 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:

s∘i=Id s \circ i = \mathrm{Id}

t∘i=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 2∈ℕk_1 \lt k_2 \in \mathbb{N} the induced structure on the 2-graph C k 2→→C k 1→→C 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).

Properties

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.

As shapes for higher categories

Globular sets may be used as a geometric shape for algebraic definitions of higher categories:

• Equipped with strictly compatible composition structure on cells in any dimension, globular sets model strict ∞-categories (originally often: “omega-categories”, see also at complicial sets), historically one of the earliest notions of higher category theory but too restrictive to be useful for most purposes (in their further restriction to strict omega-groupoids they are equivalent just to crossed complexes).

• A more general (semi-strict) notion of $n$-categories modeled on globular sets are known as associative $n$-categories, see there for more, and see a corresponding proof assistant: homotopy.io (previously: Globular . ).

• graph

• directed graph

  • directed nn-graph

  • directed $(n,r)$-graph

• ribbon graph

• Theta category,

• globular theory, globular operad

• Also related is the notion of computad, which is similar to a globular set in some ways, but allows “formal composites” of nn-cells to appear in the sources and targets of (n+1)(n+1)-cells.

• simplicial object

  • simplicial set

  • simplicial object in an (∞,1)-category

• semi-simplicial object

  • semisimplicial set

• strict infinity-category, strict omgega-groupoid?strict omega-groupoid

• associative n-category, Globular

References

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

• Tom Leinster: Higher operads, higher categories (arXiv:math/0305049)

See also

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

• R. Street, The petit topos of globular sets , JPAA 154 (2000) pp.299-315.

• Denis-Charles Cisinski, Les préfaisceaux comme types d’homotopie, Astérisque 308 Soc. Math. France (2006), 392 pages [[numdam:AST_2006__308__R1_0](http://www.numdam.org/item/?id=AST_2006__308__R1_0) pdf]

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 complexesPhilip J. Higgins, The equivalence of ∞\infty-groupoids and crossed complexes, Cah. Top. G'eom. Diff. 22 (1981) 371-386.