ترغب بنشر مسار تعليمي؟ اضغط هنا

Yoneda Lemma for $mathcal{D}$-Simplicial Spaces

121   0   0.0 ( 0 )
 نشر من قبل Nima Rasekh
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English
 تأليف Nima Rasekh




اسأل ChatGPT حول البحث

For a small category $mathcal{D}$ we define fibrations of simplicial presheaves on the category $mathcal{D}timesDelta$, which we call localized $mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(infty,n)$-categories, for models of $(infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(infty,n)$-categories.



قيم البحث

اقرأ أيضاً

137 - V. Hinich 2015
We present a version of enriched Yoneda lemma for conventional (not infinity-) categories. We require the base monoidal category to have colimits, but do not require it to be closed or symmetric monoidal.
169 - V. Hinich 2018
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected. Version 3: a few minor corrections. Version 4: Section 8 added, describing correspondences of enriched categories. In case the basic monoidal category M is a prototopos with a cartesian structure, we prove that the category of correspondences is equivalent to the category of enriched categories over [1]. Version 5: terminology changed (former bicartesian fibrations became bifibrations), a few misprints corrected. Version 6: Section 2.11 added, dealing with operadic sieves. A number of corrections and clarifications made per referees request. Version 7: final version, accepted to Advances in Math. Version 8: a minor correction of 2.8.9-2.8.10.
143 - Louis Martini 2021
We develop some basic concepts in the theory of higher categories internal to an arbitrary $infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yonedas lemma for internal categories.
126 - Seunghun Lee 2013
We prove the uniqueness, the functoriality and the naturality of cylinder objects and path objects in closed simplicial model categories.
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of G{a}lvez, Kock, and Tonks, are characterized by the property of sending ce rtain commuting squares in the simplex category $Delta$ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in $Delta$ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in $Delta$, which we characterize completely, along with several other classes of squares in $Delta$. Examples of simplicial sets with completeness conditions include quasicategories, Kan complexes, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example which we discuss in a companion paper.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا