homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
Classically, models of weak -categories comprise sets of cells in dimension 0 up to . 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, -fold categories, defined by iterated internalization
The category of strict -categories is defined by iterated enrichment:
There is an embedding
such that a strict -category is an -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 -category.
For instance, in the case , 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 -fold categories, to model weak -categories. There are strict embeddings
so that the category of weakly globular -fold categories is intermediate between strict -categories and -fold categories.
In the case , a weakly globular double category is a double category satisfying two conditions: the first is the weak globularity condition, stating that the category of objects and vertical arrows is equivalent to a discrete category, that is it is an equivalence relation. The set of connected components of 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 , a weakly globular -fold category is an -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 , are now instead only equivalent to sets. The precise notion we use for these substructures is the one of homotopically discrete -fold categories. The latter are defined inductively, starting with equivalence relations in the case of . When , the idea of a homotopically discrete -fold category is that it is an -fold category suitably equivalent to a discrete one both ‘globally’ and in each simplicial dimension.
The additional conditions in the definition of 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 -fold categories have been shown to satisfy the homotopy hypothesis and to be suitably equivalent to the Tamsamani-Simpson model.
The definition of weakly globular -fold category requires a preliminary notion, the one of homotopically discrete -fold category.
Let . Suppose, inductively, we have defined the subcategory of homotopically discrete -fold categories. We say that the -fold category is homotopically discrete if:
a) is a levelwise equivalence relation, that is for each , is an equivalence relation (that is, a category equivalent to a discrete one).
b) where with the isomorphism classes of objects functor.
When we denote by the subcategory of consisting of equivalence relations.
Let . Denote by the morphism given by
Denote by
and by the composite
We call the discretization map.
Given , for each denote by the fiber at of the map
should be thought of as a hom--category.
Define inductively -equivalences in . For , a 1-equivalence is an equivalence of categories. Suppose we have defined -equivalences in . Then a map in is an -equivalence if
a) For all ,
is an -equivalence.
b) is an -equivalence.
The next proposition means that homotopically discrete -fold categories are a higher categorical ‘fattening’ of sets.
Let . Then the maps and are -equivalences.
There are three Segal-type models of weak -categories Here is the Tamsamani-Simpson model and (called weakly globular Tamsamani -categories) is a generalization of it using weak globularity; the latter contains weakly globular -fold categories as special case.
The definition of is by induction on , starting with and 1-equivalences being equivalences of categories. Suppose, inductively, that we defined and -equivalences. Then we define through the following conditions:
a) There is an embedding of into functor categories
b) There is a truncation functor
for each and , where is the isomorphism classes of objects functor.
c) is a homotopically discrete -fold category. This comes with a -equivalence where is a discrete -fold category.
d) For each the induced Segal maps
are -equivalences in . The maps arise from the commutativity of the diagram
e) To complete the inductive step, we define -equivalences in . For this, given and , we let be the fiber at of the map
We define a map in to be an -equivalence if the following conditions hold:
i) For all
is a -equivalence.
ii) is a -equivalence.
The category of Tamsamani -categories is the full subcategory of whose objects are such that and are discrete for all , .
The embedding (1) restricts to an embedding
and the functor in (2) restricts to
The induced Segal maps coincide with the Segal maps
and are -equivalences in .
Let . Having defined the full subcategory , the category of weakly globular -fold categories is the full subcategory of whose objects are such that , for all and .
a) There is a functor ‘rigidification’
and for each an -equivalence in natural in
b) There is a functor ‘discretization’
and, for each , a zig-zag of -equivalences in between and .
The functors discretization and rigidification are used in the comparison result between Tamsamani -categories and weakly globular -fold categories as follows:
The functors
induce an equivalence of categories after localization with respect to the -equivalences
There is a subcategory of groupoidal weakly globular -fold categories which is an algebraic model of homotopy -types. This means that weakly globular -fold categories satisfy the homotopy hypothesis. To define we first consider the groupoidal version of the largest of the three Segal-type models:
The full subcategory of groupoidal weakly globular Tamsamani -categories is defined inductively as follows.
For , is the category of groupoids. Note that . Suppose inductively we have defined . We define such that
i) for all .
ii) .
From this definition it is immediate that homotopically discrete -fold categories are a full subcategory of .
The category of groupoidal weakly globular -fold categories is the full subcategory of whose objects are in .
The category of groupoidal Tamsamani -categories is the full subcategory of whose objects are in .
The functors
induce an equivalence of categories after localization with respect to the -equivalences
Since Tamsamani -groupoids are a model of -types, as a consequence of the previous proposition we obtain that weakly globular -fold categories are an algebraic model of -types.
There is an alternative more explicit way to obtain a fundamental functor
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 is given by the composite
Here is the singular functor and is the functor induced by ordinal sum , that is
The functor is left adjoint to the -fold nerve functor . While the computation of is in general very difficult, this becomes easy on those multi-simplicial sets in the essential image of the functor and one obtain
where denotes the fundamental groupoid functor in the direction. Using this expression of one can check that . In conclusion
The main reference for the theory of weakly globular -fold categories is the following research monograph, which contains an account of all three Segal-type models
The case was originally introduced in the following paper
The theory or Tamsamani weak -categories was originally developed in
Tamsamani weak -categories were further studied (from a model category theoretic perspective) in
The groupoidal case of weakly globular -fold structures started in
An even earlier precursor in the groupoidal case, restricted to modelling path connected -types and related to the -groups model, can be found in
Last revised on January 2, 2023 at 20:04:04. See the history of this page for a list of all contributions to it.