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We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $\Theta_n$-spaces and Verity's $n$-complicial sets.

We show that any pasting diagram in any(∞,2)(\infty ,2)-category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a pasting scheme is the homotopy colimit of its cells as an(∞,2)(\infty ,2)-category. We prove this explicitly in the simplicial categories model and then explain how to deduce the model-independent statement from that calculation.

Strong Steiner $\omega$-categories are a class of $\omega$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner $\omega$-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the $\omega$-categorical intuition. The purpose of this paper is to characterize this class as the class of polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of $\omega$-categories., Comment: 27 pages, v2: final version

In previous work, we develop a generalized Waldhausen $S_{\bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{\bullet}$-constructions for exact categories and for stable and exact $(\infty,1)$-categories, as well as the relative $S_{\bullet}$-construction for exact functors.

We consider the decalage construction $${{\,\mathrm{Dec}\,}}$$ and its right adjoint $$T$$. These functors are induced on the category of simplicial objects valued in any bicomplete category $${\mathcal {C}}$$ by the ordinal sum. We identify $$T{{\,\mathrm{Dec}\,}}X$$ with the path object $$X^{\Delta [1]}$$ for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit $$X\rightarrow T{{\,\mathrm{Dec}\,}}X$$ of the adjunction $$({{\,\mathrm{Dec}\,}},T)$$. When $${\mathcal {C}}$$ is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in $${\mathcal {C}}$$.

We show that the edgewise subdivision of a 2 2 -Segal object is always a Segal object, and furthermore that this property characterizes 2 2 -Segal objects.

We show that the canonical homotopy theory of strict 2-categories embeds in that of ( ∞ , 2 ) -categories in the form of 2-(pre)complicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between the canonical model structure for 2-categories and the model structure for 2-precomplicial sets. Furthermore, we show that the former model structure is transferred along this nerve and that the nerve is homotopically fully faithful.

We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models., Final version

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

We give an explicit and purely combinatorial description of the Duskin nerve of any ( r + 1 ) -point suspension 2-category, and in particular of any 2-category belonging to Joyal's cell category Θ 2 .

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.

It is known by results of Dyckerhoff-Kapranov and of G\'alvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras., Comment: 48 pages. Final version. Will appear in Proceedings of WIT

Rivista Italiana di Filosofia Analitica Junior, Vol 4, No 2 (2013): Epistemology

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. By plugging in the correct parameters, we recover several classical theorems. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group., Comment: 36 pages; Revised exposition, main results unchanged