Simplicial methods for higher categories. Segal-type models of weak \(n\)-categories
\textbf{The setting.} The appropriate notion of weak \(n\)-category has been a subject of forefront research in the recent years. Various authors in the works \begin{itemize} \item[-] \textit{R. Street}, J. Pure Appl. Algebra 49, 283--335 (1987; Zbl 0661.18005) \item[-] \textit{A. Joyal}, ``Disks, duality and \(\Theta\)-categories'', unpublished preprint (1997) \item[-] \textit{M. A. Batanin}, Adv. Math. 136, No. 1, 39--103 (1998; Zbl 0912.18006) \item[-] \textit{J. C. Baez} and \textit{J. Dolan}, Adv. Math. 135, No. 2, 145--206 (1998; Zbl 0909.18006) and \item[] \textit{C. Hermida}, \textit{M. Makkai} and \textit{J. Power}, J. Pure Appl. Algebra 154, No. 1--3, 221--246 (2000; Zbl 0971.18005); ibid. 157, No. 2--3, 247--277 (2001; Zbl 0985.18006) and ibid. 166, No. 1--2, 83--104 (2002; Zbl 0992.18005) \item[-] \textit{J. Penon}, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31--80 (1999; Zbl 0918.18006) \item[-] \textit{Z. Tamsamani}, \(K\)-Theory 16, No. 1, 51--99 (1999; Zbl 0934.18008) and \item[] \textit{C. Simpson}, Homotopy theory of higher categories. From Segal categories to \(n\)-categories and beyond. Cambridge: Cambridge University Press (2012; Zbl 1232.18001) \item[-] \textit{T. Trimble,} whose ideas were published first in [\textit{T. Leinster}, Theory Appl. Categ. 10, 1--70 (2002; Zbl 0987.18007)] \item[-] \textit{T. Leinster}, Theory Appl. Categ. 12, 73--194 (2004; Zbl 1065.18006) \item[-] \textit{J. Kock}, IMRP, Int. Math. Res. Pap. 2006, No. 1, 54 p. (2006; Zbl 1158.18008) \end{itemize} proposed different definitions that were motivated by, and applied in different situations. In the research monograph under review not only a basically new variant and its study can be found but also a principally new approach to the subject. This is based on the particularly simple structure of iterated internal categories (that is, \(n\)-fold categories). One of the main results of the book is that the new approach is homotopically equivalent to the Tamsamani-Simpson model above. \textbf{Presentation.} The book is written in a way providing the reader with great help in navigating its technical complexity. It is signposted by road maps that explain the inter-relations between concepts. It also contains informal discussions which make the ideas accessible and convey the big picture. Geometric representations in dimensions 2 and 3 are given. The book contains a thorough introduction to multisimplicial techniques making it reasonably self-contained. \textbf{The new paradigm.} For a clear explanation of the key idea, \textit{strict \(n\)-categories} are revisited first. Strict \(n\)-categories are defined iteratively. The category of strict \(0\)-categories is \(0\text{-}\mathsf{cat}:=\mathsf{set}\), the category of sets. For positive integers \(n\), the category \(n\text{-}\mathsf{cat}\) of strict \(n\)-categories is is defined as the category of categories enriched in \((n-1)\text{-}\mathsf{cat}\). Via the fully faithful nerve functor, the category \(\mathsf{cat}(\mathcal V)\) of internal categories in some category \(\mathcal V\) is embedded into the functor category \([\Delta^{\mathsf{op}}, \mathcal V]\) of simplicial objects in \(\mathcal V\) (where \(\Delta\) is the algebraists' simplicial category). A simplicial object \(\Delta^{\mathsf{op}}\to \mathcal V\) belongs to the essential image of \(\mathsf{cat}(\mathcal V)\) if and only if its higher simplices encode a strictly associative and unital composition. This can be equivalently reformulated by requiring that the Segal maps --- which send \(k\)-simplices to the length \(k\)-strings of adjacent 1-simplices along their edges --- be isomorphisms for all levels \(k\). The relation between enriched and internal categories in a nice enough category \(\mathcal V\) was studied in [\textit{A. Ehresmann} and \textit{C. Ehresmann}, Cah. Topol. Géom. Différ. 19, 387--443 (1978; Zbl 0415.18006)]. By their findings, \(n\text{-}\mathsf{cat}\) is equivalent to a full subcategory of \(\mathsf{cat}((n-1)\text{-}\mathsf{cat})\). An internal category \(X\) in \((n-1)\text{-}\mathsf{cat}\) belongs to the essential image of \(n\text{-}\mathsf{cat}\) if and only if the object of objects \(X_0\) is a discrete \((n-1)\)-category. Iteration of this embedding \(n\text{-}\mathsf{cat}\to \mathsf{cat} ((n-1)\text{-}\mathsf{cat})\) and the nerve functor \(\mathsf{cat} ((n-1)\text{-}\mathsf{cat}) \to [\Delta^{\mathsf{op}},(n-1)\text{-}\mathsf{cat}]\) yields an embedding \(n\text{-}\mathsf{cat} \to \mathsf{cat}^n \to [(\Delta^{n-1})^{\mathsf{op}}, \mathsf{cat}]\) where \(\mathsf{cat}^0=\mathsf{set}\) and for all positive integers \(n\), \(\mathsf{cat}^n=\mathsf{cat} (\mathsf{cat}^{n-1})\) is the iteratively defined category of \textit{\(n\)-fold categories}. The functor category \([(\Delta^{n-1})^{\mathsf{op}}, \mathsf{cat}]\) is the category of \textit{\((n-1)\)-simplicial categories}. Combining the results of the previous two paragraphs, an \((n-1)\)-simplicial category \(X\) belongs to the essential image of \(n\text{-}\mathsf{cat}\) if and only if \begin{itemize} \item[{(i)}] \(X_{k_1\dots k_r0}\in [(\Delta^{n-2-r})^{\mathsf{op}}, \mathsf{cat}]\) is a constant functor taking values in some discrete category, for all \(0\leq r \leq n-2\) and all \(k_1\dots k_r\in \Delta^{r}\), and \item[{(ii)}] the Segal maps of \(X\) are isomorphisms for all levels in all of the \(n-1\) directions. \end{itemize} Condition (i) is termed the \textit{globularity condition} because it says that certain boundaries of any simplex are trivial; that is, all simplices have a globular shape. Condition (ii) is referred to as the \textit{Segal condition}. It is equivalent to \(X\) being the multinerve of an \(n\)-fold category; that is, the composition encoded in \(X\) being strictly associative and unital in every direction. In order to generalize strict \(n\)-categories to weak ones, one can naturally weaken the globularity condition (i) or the Segal condition (ii). The second possibility was taken in Tamsamani's above cited work. In a Tamsamani weak \(n\)-category, the Segal maps are allowed to be \((n-1)\)-equivalences rather than isomorphisms; but the globularity condition is required in the same form above. As the groundbreaking new concept in the book under review, a suitable weakening of the globularity condition (i) is proposed. Generalizing discrete categories occurring in the globularity condition, \textit{homotopically discrete categories} are introduced. A \textit{weakly globular \(n\)-fold category} is defined as an \(n\)-fold category \(X\) satisfying, among others, the weak globularity condition \begin{itemize} \item[] \(X_{k_1\dots k_r0}\in [(\Delta^{n-2-r})^{\mathsf{op}}, \mathsf{cat}]\) are constant functors taking values in some homotopically discrete category, for all \(0\leq r \leq n-2\) and all \(k_1\dots k_r\in \Delta^{r}\). \end{itemize} The requirement that \(X\) is an \(n\)-fold category means that the Segal condition is still required in the strict sense that the Segal maps are isomorphisms. All that is placed in the broader framework of \textit{weakly globular Tamsamani \(n\)-categories} where both conditions (i) and (ii) are weakened: the underlying structure only has to be homotopically discrete and the (accordingly modified) Segal maps only need to be \((n-1)\)-equivalences. This results in the following picture of subcategories. \[ \begin{tikzcd} & n \text{-} \mathsf{cat} \ar[ld, hook] \ar[rd, hook'] \\ \mathsf{ta}^n \ar[rd, hook] & & \mathsf{cat}^n_{\mathsf{wg}} \ar[ld, hook'] \\ & \mathsf{ta}^n_{\mathsf{wg}} \end{tikzcd} \] The picture is made complete by the comparison of the category \(\mathsf{ta}^n\) of Tamsamani \(n\)-categories and the category \(\mathsf{cat}^n_{\mathsf{wg}}\) of weakly globular \(n\)-fold categories: their localizations with respect to \(n\)-equivalences are proven to be equivalent. That is to say, from a homotopy theoretical point of view, Tamsamani's weak \(n\)-categories and Paoli's weakly globular \(n\)-fold categories offer equivalent approaches. Consequently, the homotopy hypothesis holds: localization with respect to \(n\)-equivalences of the full subcategory of groupoidal objects in \(\mathsf{cat}^n_{\mathsf{wg}}\) is equivalent to the category of homotopy \(n\)-types. \textbf{Homotopically discrete \(n\)-fold categories.} The novel approach in the reviewed work is based on the replacement of discrete categories in the globularity condition (i) above with a suitable generalization named \textit{homotopically discrete \(n\)-fold categories}. For \(n=0\), \(\mathsf{cat}^0_{\mathsf{hd}}:=\mathsf{set}\) and for all positive integers \(n\), the full subcategory \(\mathsf{cat}^n_{\mathsf{hd}}\subseteq \mathsf{cat}^n\) of homotopically discrete \(n\)-fold categories is defined iteratively. Namely, an \(n\)-fold category is homotopically discrete if \begin{itemize} \item[-] its multinerve \((\Delta^n)^{\mathsf{op}}\to \mathsf{set}\) is, at each level, the nerve of a groupoid without non-trivial loops (that is, of an equivalence relation), and \item[-] \((n-1)\)-truncation turns it to a homotopically discrete \((n-1)\)-fold category. \end{itemize} Homotopically discrete \(n\)-fold categories can be seen as iterated internal equivalence relations. In particular, \(\mathsf{cat}^1_{\mathsf{hd}}\) is the category of equivalence relations; that is, the category of groupoids without non-trivial loops. Every homotopically discrete \(n\)-fold category comes together with its discretized version and a higher categorical equivalence --- the discretization map --- between them. Groupoidal homotopically discrete \(n\)-fold categories represent 0-types. \textbf{Weakly globular Tamsamani \(n\)-categories.} They are defined iteratively as simplicial objects in the category of weakly globular Tamsamani \((n-1)\)-categories with two additional properties: \begin{itemize} \item[{(i')}] \(X_0\) is a homotopically discrete \((n-1)\)-fold category and \item[{(ii')}] the (modified) Segal maps are \((n-1)\)-equivalences. \end{itemize} In order to see that the modified Segal maps are well-defined, some closedness properties of the category \(\mathsf{ta}^{n}_{\mathsf{wg}}\) of weakly globular Tamsamani \(n\)-categories are verified also using closedness properties of \(\mathsf{cat}^n_{\mathsf{hd}}\). The \(n\)-equivalences in \(\mathsf{ta}^{n}_{\mathsf{wg}}\) are identified; they are the higher dimensional analogue of a fully faithful functor which is essentially surjective on the objects. \textit{Tamsamani \(n\)-categories} are re-obtained as weakly globular Tamsamani \(n\)-categories satisfying the globularity condition in the strict form (i). Under this condition the modified Segal maps reduce to the original Segal maps of (ii). \textbf{Weakly globular \(n\)-fold categories.} They are weakly globular Tamsamani \(n\)-categories whose \(r\)-truncation is an \(r\)-fold category for all \(r\leq n\). This definition leads to a more explicit characterization: an \(n\)-fold category is weakly globular if and only if the following iteratively formulated conditions hold for its nerve \(X:\Delta^{\mathsf{op}} \to \mathsf{cat}^{n-1}\). \begin{itemize} \item[-] \(X_0\) is a homotopically discrete \((n-1)\)-fold category, \item[-] for all positive integers \(k\), \(X_k\) is a weakly globular \((n-1)\)-fold category, \item[-] the original and the modified Segal maps differ by \((n-1)\)-equivalences and \item[-] the \((n-1)\)-truncation of \(X\) gives a weakly globular \((n-1)\)-fold category. \end{itemize} \textbf{Mutually inverse equivalences and the homotopy hypothesis.} First a functor \(Q_n:\mathsf{ta}^n_{\mathsf{wg}} \to \mathsf{cat}^n_{\mathsf{wg}}\) is constructed, together with a natural transformation from \( \begin{tikzcd} \mathsf{ta}^n_{\mathsf{wg}} \ar[r, "Q_n"] & \mathsf{cat}^n_{\mathsf{wg}} \ar[r, hook] & \mathsf{ta}^n_{\mathsf{wg}} \end{tikzcd} \) to the identity functor, whose components are \(n\)-equivalences. This functor \(Q_n\) arises as a composite of three ingredients. An `approximation up-to \(n\)-equivalence' functor is constructed from \(\mathsf{ta}^n_{\mathsf{wg}}\) to a suitable subcategory of \(\mathsf{ta}^n_{\mathsf{wg}}\); chosen in such a way that it admits a `transport of structure' functor to the category \(\mathsf{segps}[(\Delta^{n-1})^{\mathsf{op}} ,\mathsf{cat}]\) of Segalic pseudofunctors \((\Delta^{n-1})^{\mathsf{op}} \to \mathsf{cat}\). Segalic pseudofunctors are distinguished pseudofunctors with additional discreteness properties which allow the Segal maps to exist, for which these Segal maps are isos, and which allow for a `strictification functor' \(\mathsf{segps}[(\Delta^{n-1})^{\mathsf{op}} ,\mathsf{cat}] \to \mathsf{cat}^n_{\mathsf{wg}}\) built using the classical strictification machinery of [\textit{A. J. Power}, J. Pure Appl. Algebra 57, No. 2, 165--173 (1989; Zbl 0668.18010)]. The functor \(Q_n\) restricts to the \textit{rigidification functor} \(\mathsf{ta}^n \to \mathsf{cat}^n_{\mathsf{wg}}\). When \(n=2\), this functor was already constructed in [\textit{S. Paoli} and \textit{D. Pronk}, Theory Appl. Categ. 28, 933--980 (2013; Zbl 1278.18006)]. However, the \(n>2\) case requires substantially new ideas and constructions which first appear here. In the opposite direction, the \textit{discretization functor} \(\mathsf{cat}^n_{\mathsf{wg}} \to \mathsf{ta}^n\) is constructed as a composite of two functors. Certain `approximation up-to equivalence' functors \(\mathsf{cat}^n_{\mathsf{wg}} \to \mathsf{cat}^n_{\mathsf{wg}}\) are defined iteratively. They induce a functor from \(\mathsf{cat}^n_{\mathsf{wg}}\) to its full subcategory \(\mathsf{fcat}^n_{\mathsf{wg}}\) whose objects have homotopically discrete substructures with functorial sections of the discretization maps. This functoriality requirement makes it possible to define a `discretization functor' \(\mathsf{fcat}^n_{\mathsf{wg}} \to \mathsf{ta}^n\). After localization with respect to \(n\)-equivalences, the rigidification and the discretization functors above induce an equivalence \(\mathsf{cat}^n_{\mathsf{wg}}/\!\!\sim_n \,\simeq \mathsf{ta}^n /\!\!\sim_n\). Since the homotopy hypothesis holds for Tamsamani weak \(n\)-categories, the above equivalence implies that it also holds for weakly globular \(n\)-fold categories. That is, after localization with respect to \(n\)-equivalences, the full subcategory of groupoidal weakly globular \(n\)-fold categories becomes equivalent to the category of homotopy \(n\)-types. It is shown that this equivalence can be realized using the particularly simple an explicit fundamental \(n\)-groupoid functor of [\textit{D. Blanc} and \textit{S. Paoli} Algebr. Geom. Topol. 14, No. 6, 3419--3491 (2014; Zbl 1311.55025)]. \textbf{Future applications.} The following future applications of the theory, developed here, are envisaged. \begin{itemize} \item[-] An algebraic understanding of \(k\)-invariants in Postnikov systems. \item[-] Construction of a Quillen model structure on \(\mathsf{cat}^n_{\mathsf{wg}}\). \item[-] A tool to compare some of the notions of weak \(n\)-category listed above. \item[-] Formulation of semistrict \(n\)-categories and a proof of the `weak units conjecture' in [\textit{C. Simpson}, ``Homotopy types of strict 3-groupoids'', Preprint, \url{arXiv:math/9810059}]. \item[-] Extension to a study of \((\infty,n)\)-categories. \end{itemize}
scientific article; zbMATH DE number 7029566