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Title
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Templicial objects : simplicial objects in a monoidal category
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Author
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Abstract
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| Based on the work of Aguiar and Leinster, we introduce templicial (short for tensor-simplicial) objects which may be viewed as simplicial objects internalized into a (non-cartesian) monoidal category V. They are defined as certain colax monoidal functors on the category of finite intervals, as opposed to functors on the usual simplex category. Templicial objects still have all degeneracy and inner face maps, but the outer face maps are replaced by the comultiplication maps. The usual nerve construction applied to an enriched category fails to provide a simplicial object, but it very naturally produces a templicial object. The better part of this thesis is devoted to constructing templicial analogues of different classical nerve constructions, most notably Cordier's homotopy coherent nerve and Lurie's dg-nerve (when V is the category of modules over a commutative ring k). The former makes essential use of Dugger and Spivak's necklaces. We show that these analogues are compatible with their classical counterparts, with which they share similar properties. Further, we show that all these templicial nerves possess natural extra data which we call (non-associative) Frobenius structures. Moreover, the templicial analogue of the dg-nerve provides an equivalence of categories between templicial k-modules with a Frobenius structure and non-negatively graded dg-categories. Finally, we introduce what we consider to be the natural analogue of Joyal's quasi-categories in the context of templicial objects, called quasi-categories in V. They satisfy similar properties to their classical counterparts and all of the templicial nerves provide examples of them. We also show that every projective quasi-category in V comes equipped with a non-associative Frobenius structure and that the converse holds in case V is the category of k-modules. The study of the homotopy theory of quasi-categories in V is left to future research but we hypothesize that they form a model for infinity-categories enriched in simplicial objects. |
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Language
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English
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Publication
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Antwerp
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University of Antwerp, Faculty of Science
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2022
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Volume/pages
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xviii, 172 p.
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Note
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:
Lowen, Wendy [Supervisor]
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Full text (open access)
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