# Categorical models for path spaces

@inproceedings{Minichiello2022CategoricalMF, title={Categorical models for path spaces}, author={Emilio Minichiello and Manuel Rivera and Mahmoud Zeinalian}, year={2022} }

We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by considering localizations of the rigidification functor, unraveling a construction of Hinich, and using a sequence of operators originally introduced by Szczarba in 1961. As a result, we obtain several combinatorial space-level models for the path category of a…

## 3 Citations

### The simplicial coalgebra of chains under three different notions of weak equivalence

- Mathematics
- 2022

We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced…

### The cobar construction as an $E_{\infty}$-bialgebra model of the based loop space

- Mathematics
- 2021

In the fifties, Adams introduced a comparison map θZ : ΩS (Z, z) → S (Ωz Z) from his cobar construction on the (simplicial) singular chains of a pointed space (Z, z) to the cubical singular chains on…

### Classifying Space via Homotopy Coherent Nerve

- Mathematics
- 2022

. We prove that the classifying space of a simplicial group is mod- eled by its homotopy coherent nerve. The proof is an amalgamation of the idea in [Hin07] and Dmitri Pavlov’s answer given in [Mat].

## References

SHOWING 1-10 OF 36 REFERENCES

### Homology and fibrations I Coalgebras, cotensor product and its derived functors

- Mathematics
- 1965

The s tudy of the relations between the homology structure of the base space, the total space and the fiber of a fibration offers ample opportunity for application of homological algebra. This series…

### Cubical rigidification, the cobar construction and the based loop space

- MathematicsAlgebraic & Geometric Topology
- 2018

We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the…

### A MODEL STRUCTURE FOR QUASI-CATEGORIES

- Mathematics
- 2012

Quasi-categories live at the intersection of homotopy theory with category theory. In particular, they serve as a model for (∞, 1)-categories, that is, weak higher categories with n-cells for each…

### Adding inverses to diagrams II: Invertible homotopy theories are spaces

- Mathematics
- 2007

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen…

### Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2022

We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing…

### THE LOOP GROUP AND THE COBAR CONSTRUCTION

- Mathematics
- 2009

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan…

### Elements of ∞-Category Theory

- Philosophy
- 2022

The language of
∞-categories provides an insightful new way of expressing many
results in higher-dimensional mathematics but can be challenging
for the uninitiated. To explain what exactly an…

### THE HOMOLOGY OF TWISTED CARTESIAN PRODUCTS

- Mathematics
- 1961

Introduction. In a recent paper [2], E. H. Brown introduced the notion of a twisted tensor product. Briefly, the definition is as follows. Let K be a D.G.A. (differential, graded, augmented)…