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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 simplicial set. We then pass to the chain-level and describe a model for the path category, now considered as a category enriched over differential graded (dg) coalgebras, in terms of a suitable algebraic chain model for the underlying space. This is achieved through a localized version of the cobar functor from the categorical Koszul duality theory of Holstein and Lazarev and considering the chains on a simplicial set as a curved coalgebra equipped with higher structure. We obtain a conceptual explanation of a result of Franz stating that the dg algebra quasi-isomorphism from the extended cobar construction on the chains of a reduced simplicial set to the chains on its Kan loop group, originally constructed by Hess and Tonks in terms of Szczarba's twisting cochain, is a map of dg bialgebras. Contents 1. Introduction 1.1. Organization of the paper 1.2. Acknowledgments 2. Quivers, simplicial sets, and localization 2.1. Quivers and simplicial sets 2.2. Localization 3. The Szczarba map 3.1. The Kan loop groupoid functor 3.2. The rigidification functor 3.3. Rigidification in terms of necklaces 3.4. The natural transformation Sz 4. Simplicial models for the path category 4.1. Comparing the classifying space and homotopy coherent nerve functors 4.2. Comparing the localized rigidification and Kan loop groupoid functors 4.3. Comparing the localized rigidification and the path category functors 5. Algebraic models for the path category 5.1. Categorical coalgebras 5.2. Normalized chains as a categorical coalgebra 5.3. The cobar functor, necklaces, and cubes 5.4. B ∞ -coalgebras 5.5. Normalized chains as a B ∞ -coalgebra 5.6. The extended cobar construction 5.7. Proof of Theorem 1.2 5.8. The extended cobar construction as a model for the path category Appendix A. Model Structures A.1. The Kan-Quillen model structure on simplicial sets A.2. The Joyal model structure on simplicial sets A.3. The Dwyer-Kan model structure on simplicial groupoids

Given a diffeological group G we prove that the nerve of the groupoid of diffeological principal G-bundles over a diffeological space X is weakly equivalent to the nerve of the category of G-principal ∞-bundles over X. Using this we show that the Čech cohomology groups of X as defined in [Igl20] coincides with the corresponding cohomology groups defined using higher topos theory. Contents 24 8. Comparison to Čech Cohomology 28 References 33

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