nLab weakly globular n-fold category



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



Classically, models of weak n n -categories comprise sets of cells in dimension 0 up to nn. This is also called the globularity condition. Geometrically, it corresponds to cells having a globular shape

On the other hand, non-globular structures exist in higher category theory: for instance, n n -fold categories, defined by iterated internalization

Cat 0=Set,Cat n=Cat(Cat n1). Cat^{0}=Set,\qquad\qquad Cat^{n}=Cat(Cat^{n-1})\;.

The category n-Catn\text{-}Cat of strict n n -categories is defined by iterated enrichment:

0-Cat=Set,n-Cat=((n1)-Cat,×)-Cat. 0\text{-}Cat=Set,\qquad\qquad n\text{-}Cat=((n-1)\text{-}Cat,\times)\text{-}Cat\;.

There is an embedding

n-CatCat n n\text{-}Cat\hookrightarrow Cat^{n}

such that a strict nn-category is an nn-fold category in which certain substructures are discrete (that is, sets). This discreteness condition is precisely the globularity condition and the sets underlying these discrete substructures are the sets of cells in the strict nn-category.

For instance, in the case n=2n=2, strict 2-categories are double categories in which the category of objects and vertical arrows is discrete. In pictures: We see that the picture on the right becomes the one on the left when all vertical morphisms are identities.

In the weakly globular approach to higher categories, the cells in each dimension instead of forming a set, have a higher categorical structure which is homotopically discrete, that is only equivalent (in a higher categorical sense) to a set. This condition, called weak globularity condition, is a new paradigm to weaken higher categorical structures and allows to use rigid structures, namely nn-fold categories, to model weak nn-categories. There are strict embeddings

n-CatCat wg nCat n n\text{-} Cat \hookrightarrow Cat_{wg}^{n} \hookrightarrow Cat^{n}

so that the category Cat wg nCat_{wg}^{n} of weakly globular nn-fold categories is intermediate between strict nn-categories and nn-fold categories.

In the case n=2n=2, a weakly globular double category XX is a double category satisfying two conditions: the first is the weak globularity condition, stating that the category X 0X_0 of objects and vertical arrows is equivalent to a discrete category, that is it is an equivalence relation. The set pX 0pX_0 of connected components of X 0X_0 plays the role of ‘set of objects’ in the weak structure. We therefore need to define a composition of horizontal arrows whose source and target are in the same vertical connected component

For this purpose, we impose a second condition: for each ‘staircase path’

A more concise way to express this second condition is in term of the so called induced Segal maps condition.

Weakly globular double categories have been shown by Paoli and Pronk to be suitably equivalent to bicategories.

In dimension n2n\geq2, a weakly globular nn-fold category XX is an nn-fold category which first of all needs to satisfy the weak globularity condition; namely those substructures that are discrete in the image of the embedding n-CatCat nn\text{-} Cat\hookrightarrow Cat^{n}, are now instead only equivalent to sets. The precise notion we use for these substructures is the one of homotopically discrete nn-fold categories. The latter are defined inductively, starting with equivalence relations in the case of n=1n=1. When n1n\geq1, the idea of a homotopically discrete nn-fold category is that it is an nn-fold category suitably equivalent to a discrete one both ‘globally’ and in each simplicial dimension.

The additional conditions in the definition of Cat wg nCat_{wg}^{n} are given inductively in terms of induced Segal maps conditions. They guarantee the existence of weakly associative and weakly unital compositions but, like in the Tamsamani-Simpson model, their coherences are not given explicitly but they are automatically encoded in the multi-simplicial combinatorics.

Weakly globular nn-fold categories have been shown to satisfy the homotopy hypothesis and to be suitably equivalent to the Tamsamani-Simpson model.

Homotopically discrete nn-fold categories

The definition of weakly globular nn-fold category requires a preliminary notion, the one of homotopically discrete nn-fold category.


Let Cat hd 0=SetCat_{hd}^{0}=Set. Suppose, inductively, we have defined the subcategory Cat hd n1Cat n1Cat_{hd}^{n-1}\subset Cat^{n-1} of homotopically discrete (n1)(n-1)-fold categories. We say that the nn-fold category XCat nX\in Cat^{n} is homotopically discrete if:

a) XX is a levelwise equivalence relation, that is for each (k 1,,k n1)Δ n1 op(k_1,\ldots,k_{n-1})\in\Delta^{{n-1}^{op}}, X k 1,,k n1CatX_{k_1,\ldots,k_{n-1}}\in Cat is an equivalence relation (that is, a category equivalent to a discrete one).

b) p (n1)XCat hd n1p^{(n-1)}X\in Cat_{hd}^{n-1} where (p (n1)X) k 1,,k n1=pX k 1,,k n1(p^{(n-1)}X)_{k_1,\ldots,k_{n-1}}=p X_{k_1,\ldots,k_{n-1}} with p:CatSetp: Cat\rightarrow Set the isomorphism classes of objects functor.

When n=1n=1 we denote by Cat hd 1=Cat hd Cat_{hd}^{1}=Cat_{hd}^{} the subcategory of Cat Cat consisting of equivalence relations.


Let XCat hd nX\in Cat_{hd}^{n}. Denote by γ X (n1):Xp (n1)X\gamma^{(n-1)}_X:X\rightarrow p^{(n-1)}X the morphism given by

(γ X (n1)) s 1...s n1:X s 1...s n1pX s 1...s n1. (\gamma^{(n-1)}_X)_{s_1...s_{n-1}} :X_{s_1...s_{n-1}} \rightarrow p X_{s_1...s_{n-1}}\;.

Denote by

X d=pp (1)...p (n1)X X^d =p p^{(1)}...p^{(n-1)}X

and by γ n\gamma_{n} the composite

Xγ (n1)p (n1)Xγ (n2)p (n2)p (n1)Xγ (0)X d. X\xrightarrow{\gamma^{(n-1)}}p^{(n-1)}X \xrightarrow{\gamma^{(n-2)}} p^{(n-2)}p^{(n-1)}X \rightarrow \cdots \xrightarrow{\gamma^{(0)}} X^d\;.

We call γ n\gamma_{n} the discretization map.


Given XCat hd nX\in Cat_{hd}^{n}, for each a,bX 0 da,b\in X_0^d denote by X(a,b)X(a,b) the fiber at (a,b)(a,b) of the map

X 1(d 0,d 1)X 0×X 0γ n×γ nX 0 d×X 0 d. X_1 \xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma_{n}\times\gamma_{n}} X_0^d\times X_0^d\;.

X(a,b)Cat hd n1X(a,b)\in Cat_{hd}^{n-1} should be thought of as a hom-(n1)(n-1)-category.


Define inductively nn-equivalences in Cat hd nCat_{hd}^{n}. For n=1n=1, a 1-equivalence is an equivalence of categories. Suppose we have defined (n1)(n-1)-equivalences in Cat hd n1Cat_{hd}^{n-1}. Then a map f:XYf:X\rightarrow Y in Cat hd nCat_{hd}^{n} is an nn-equivalence if

a) For all a,bX 0 da,b \in X_0^d,

f(a,b):X(a,b)Y(fa,fb) f(a,b):X(a,b) \rightarrow Y(f a,f b)

is an (n1)(n-1)-equivalence.

b) p (n1)fp^{(n-1)}f is an (n1)(n-1)-equivalence.

The next proposition means that homotopically discrete nn-fold categories are a higher categorical ‘fattening’ of sets.


Let XCat hd nX\in Cat_{hd}^{n}. Then the maps γ (n1):Xp (n1)X\gamma^{(n-1)}:X\rightarrow p^{(n-1)}X and γ n:XX d\gamma_{n}:X\rightarrow X^d are nn-equivalences.

The three Segal-type models

There are three Segal-type models of weak nn-categories Here Ta nTa^{n} is the Tamsamani-Simpson model and Ta wg nTa_{wg}^{n} (called weakly globular Tamsamani nn-categories) is a generalization of it using weak globularity; the latter contains weakly globular nn-fold categories as special case.

Weakly globular Tamsamani n{n}-categories

The definition of Ta wg nTa_{wg}^{n} is by induction on nn, starting with Ta wg 1=CatTa_{wg}^{1}= Cat and 1-equivalences being equivalences of categories. Suppose, inductively, that we defined Ta wg n1Ta_{wg}^{n-1} and (n1)(n-1)-equivalences. Then we define Ta wg nTa_{wg}^{n} through the following conditions:

a) There is an embedding of Ta wg nTa_{wg}^{n} into functor categories

(1)Ta wg n[Δ op,Ta wg n1][Δ n1 op,Cat] Ta_{wg}^{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{wg}^{n-1}}]\hookrightarrow [\Delta^{{n-1}^{op}}, Cat]

b) There is a truncation functor

(2)p (n1):Ta wg nTa wg n1 p^{(n-1)}:Ta_{wg}^{n}\rightarrow Ta_{wg}^{n-1}
(p (n1)X) k 1,,k n1=pX k 1,,k n1 (p^{(n-1)}X)_{k_1,\ldots, k_{n-1}}=p X_{k_1,\ldots, k_{n-1}}

for each (k 1,,k n1)Δ n1 op(k_1,\ldots, k_{n-1})\in\Delta^{{n-1}^{op}} and XTa wg nX\in Ta_{wg}^{n}, where p:CatSetp: Cat\rightarrow Set is the isomorphism classes of objects functor.

c) X 0X_0 is a homotopically discrete (n1)(n-1)-fold category. This comes with a (n1)(n-1)-equivalence γ:X 0X 0 d\gamma:X_0\rightarrow X_0^d where X 0 dX_0^d is a discrete (n1)(n-1)-fold category.

d) For each k2k\geq 2 the induced Segal maps

(3)μ^ k:X kX 1× X 0 dk× X 0 dX 1 \hat{\mu}_k:X_k\rightarrow {X_1\times_{X_0^d}\overset{k}{\cdots}\times_{X_0^d}X_1}

are (n1)(n-1)-equivalences in Ta wg n1Ta_{wg}^{n-1}. The maps μ^ k\hat{\mu}_k arise from the commutativity of the diagram

e) To complete the inductive step, we define nn-equivalences in Ta wg nTa_{wg}^{n}. For this, given XTa wg nX\in Ta_{wg}^{n} and (a,b)X 0 d×X 0 d(a,b)\in X_0^d\times X_0^d, we let X(a,b)X 1X(a,b)\subset X_1 be the fiber at (a,b)(a,b) of the map

X 1(d 0,d 1)X 0×X 0γ×γX 0 d×X 0 d. X_1\xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma\times\gamma} X^d_0\times X^d_0\;.

We define a map f:XYf:X\rightarrow Y in Ta wg nTa_{wg}^{n} to be an nn-equivalence if the following conditions hold:

i) For all a,bX 0 da,b\in X_0^d

f(a,b):X(a,b)Y(fa,fb) f(a,b): X(a,b)\rightarrow Y(f a,f b)

is a (n1)(n-1)-equivalence.

ii) p (n1)fp^{(n-1)}f is a (n1)(n-1)-equivalence.

Tamsamani n{n}-categories

The category Ta nTa^{n} of Tamsamani nn-categories is the full subcategory of Ta wg nTa_{wg}^{n} whose objects XX are such that X 0X_0 and X k 1k r0X_{k_1\ldots k_r 0} are discrete for all (k 1k r)Δ r op(k_1\ldots k_r)\in \Delta^{{r}^{op}}, 1rn21\leq r\leq n-2.

The embedding (1) restricts to an embedding

Ta n[Δ op,Ta n1] Ta_{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{n-1}}]

and the functor p (n1)p^{(n-1)} in (2) restricts to

p (n1):Ta nTa n1 p^{(n-1)}:Ta^{n}\rightarrow Ta^{n-1}

The induced Segal maps μ^ k\hat{\mu}_k coincide with the Segal maps

X kX 1× X 0k× X 0X 1 X_k\rightarrow {X_1\times_{X_0}\overset{k}{\cdots}\times_{X_0}X_1}

and are (n1)(n-1)-equivalences in Ta n1Ta^{n-1}.

Weakly globular nn-fold categories

Let Cat wg 1=CatCat_{wg}^{1}= Cat. Having defined the full subcategory Cat wg n1Ta wg n1Cat_{wg}^{n-1}\subset Ta_{wg}^{n-1}, the category Cat wg nCat_{wg}^{n} of weakly globular nn-fold categories is the full subcategory of Ta wg nTa_{wg}^{n} whose objects XX are such that XCat nX\in Cat^{n}, X kCat wg n1X_k\in Cat_{wg}^{n-1} for all k0k\geq 0 and p (n1)XCat wg n1p^{(n-1)}X\in Cat_{wg}^{n-1}.

Main results


a) There is a functor ‘rigidification’

Q n:Ta wg nCat wg n Q_n: Ta_{wg}^{n}\rightarrow Cat_{wg}^{n}

and for each XTa wg nX\in Ta_{wg}^{n} an nn-equivalence in Ta wg nTa_{wg}^{n} natural in XX

s n(X):Q nXX. s_n(X):Q_n X\rightarrow X.

b) There is a functor ‘discretization’

Disc n:Cat wg nTa n Disc_{n}:Cat_{wg}^{n}\rightarrow Ta^{n}

and, for each XCat wg nX\in Cat_{wg}^{n}, a zig-zag of nn-equivalences in Ta wg nTa_{wg}^{n} between XX and Disc nXDisc_{n} X.

The functors discretization and rigidification are used in the comparison result between Tamsamani nn-categories and weakly globular nn-fold categories as follows:


The functors

Q n:Ta nCat wg n:Disc n Q_n:Ta^{n}\leftrightarrows Cat_{wg}^{n}:Disc_{n}

induce an equivalence of categories after localization with respect to the nn-equivalences

Ta n/ nCat wg n/ n. Ta^{n}/\!\!\sim^n\;\simeq \; Cat_{wg}^{n}/\!\!\sim^n\;.

The groupoidal case

There is a subcategory GCat wg nCat wg nGCat_{wg}^{n}\subset Cat_{wg}^{n} of groupoidal weakly globular nn-fold categories which is an algebraic model of homotopy n n -types. This means that weakly globular nn-fold categories satisfy the homotopy hypothesis. To define GCat wg nGCat_{wg}^{n} we first consider the groupoidal version of the largest of the three Segal-type models:


The full subcategory GTa wg nTa wg nGTa_{wg}^{n}\subset Ta_{wg}^{n} of groupoidal weakly globular Tamsamani nn-categories is defined inductively as follows.

For n=1n=1, GTa wg 1=GpdGTa_{wg}^{1}=Gpd is the category of groupoids. Note that Cat hd GTa wg 1Cat_{hd}^{}\subset GTa_{wg}^{1}. Suppose inductively we have defined GTa wg n1Ta wg n1GTa_{wg}^{n-1}\subset Ta_{wg}^{n-1}. We define XGTa wg nTa wg nX\in GTa_{wg}^{n}\subset Ta_{wg}^{n} such that

i) X kGTa wg n1X_k\in GTa_{wg}^{n-1} for all k0k\geq 0.

ii) p (n1)XGTa wg n1p^{(n-1)}X\in GTa_{wg}^{n-1}.

From this definition it is immediate that homotopically discrete nn-fold categories are a full subcategory of GTa wg nGTa_{wg}^{n}.


The category GCat wg nCat wg nGCat_{wg}^{n}\subset Cat_{wg}^{n} of groupoidal weakly globular nn-fold categories is the full subcategory of Cat wg nCat_{wg}^{n} whose objects XX are in GTa wg nGTa_{wg}^{n}.

The category GTa nTa nGTa^{n}\subset Ta^{n} of groupoidal Tamsamani nn-categories is the full subcategory of Ta nTa^{n} whose objects XX are in GTa wg nGTa_{wg}^{n}.


The functors

Q n:GTa nGCat wg n:Disc n Q_n:GTa^{n}\leftrightarrows GCat_{wg}^{n}:Disc_{n}

induce an equivalence of categories after localization with respect to the nn-equivalences

GTa n/ nGCat wg n/ n. GTa^{n}/\!\!\sim^n \;\simeq\; GCat_{wg}^{n}/\!\!\sim^n\;.

Since Tamsamani nn-groupoids are a model of nn-types, as a consequence of the previous proposition we obtain that weakly globular nn-fold categories are an algebraic model of nn-types.

There is an alternative more explicit way to obtain a fundamental functor

n:n-typesGCat wg n \mathcal{H}_{n}:n\text{-types}\rightarrow GCat_{wg}^{n}

based on the work of Blanc and Paoli. This work shows in particular how the notion of weak globularity arises naturally in topology.

The functor n\mathcal{H}_n is given by the composite

n:n-types𝒮[Δ op,Set]Or n[Δ n op,Set]𝒫 nGpd n. \mathcal{H}_n: n\text{-types}\xrightarrow{\mathcal{S}} [\Delta^{{}^{op}},Set] \xrightarrow{Or_{n}} [\Delta^{{n}^{op}},Set]\xrightarrow{\mathcal{P}_n} Gpd^{n}\;.

Here 𝒮\mathcal{S} is the singular functor and Or nOr_{n} is the functor induced by ordinal sum or n:Δ n opΔor_{n}: \Delta^{{n}^{op}} \rightarrow \Delta, that is

(Or nX) p 1p n=X n1+p 1++p n (Or_{n} X)_{p_1\ldots p_n}=X_{n-1+p_1+\ldots+ p_n}

The functor 𝒫 n\mathcal{P}_n is left adjoint to the nn-fold nerve functor N n:Gpd n[Δ n op,Set]N_n:Gpd^{n}\rightarrow [\Delta^{{n}^{op}},Set]. While the computation of 𝒫 n\mathcal{P}_n is in general very difficult, this becomes easy on those multi-simplicial sets in the essential image of the functor Or n𝒮Or_{n}\mathcal{S} and one obtain

nX=π^ 1π^ 2π^ nOr n𝒮X. \mathcal{H}_n X=\hat{\pi}^{1} \hat{\pi}^{2} \cdots \hat{\pi}^{n} Or_{n} \mathcal{S} X\;.

where π^ (i)\hat{\pi}^{(i)} denotes the fundamental groupoid functor in the i thi^{th} direction. Using this expression of nX\mathcal{H}_n X one can check that nXGCat wg n\mathcal{H}_n X\in GCat_{wg}^{n}. In conclusion


The functors

BDisc n:GCat wg nn-types: n B\circ Disc_{n}:GCat_{wg}^{n}\rightleftarrows n\text{-types}:\mathcal{H}_n

induce an equivalence of categories

GCat wg n/ no(n-types). GCat_{wg}^{n}/\!\!\sim^{n}\,\simeq \,\mathcal{H}\!o(n\text{-types})\;.

where o(n-types)\mathcal{H}\!o(n\text{-types}) is the homotopy category of n-typesn\text{-types}.


The main reference for the theory of weakly globular nn-fold categories is the following research monograph, which contains an account of all three Segal-type models

  • S. Paoli. Simplicial Methods for Higher Categories: Segal-type models of weak n-categories, volume 26 of Algebra and Applications. Springer, 2019 toc pdf.

The case n=2n=2 was originally introduced in the following paper

  • S. Paoli, D. Pronk, A double categorical model of weak 2-categories, Theory and Application of categories, 28, (2013), 933-980.

The theory or Tamsamani weak nn-categories was originally developed in

  • Z. Tamsamani. Sur des notions de nn-categorie et nn-groupoide non-strictes via des ensembles multisimpliciaux. K-theory, 16:51–99, 1999.

Tamsamani weak nn-categories were further studied (from a model category theoretic perspective) in

  • C. Simpson. Homotopy theory of higher categories, volume 19 of New Math. Monographs. Cambridge University Press, 2012.

The groupoidal case of weakly globular nn-fold structures started in

  • D. Blanc and S. Paoli. Segal-type algebraic models of nn-types. Algebr. Geom. Topol., 14:3419–3491, 2014.

An even earlier precursor in the groupoidal case, restricted to modelling path connected nn-types and related to the cat n cat_n -groups model, can be found in

  • S. Paoli. Weakly globular cat ncat_n-groups and Tamsamani’s model. Adv. in Math., 222:621–727, 2009.

Last revised on January 2, 2023 at 20:04:04. See the history of this page for a list of all contributions to it.