Segal objects and the Grothendieck construction

2018 ◽  
pp. 19-44
Author(s):  
Pedro Boavida de Brito
Keyword(s):  
Grothendieck Construction

scholarly journals The Grothendieck construction for model categories

2015 ◽  
Vol 281 ◽  
pp. 1306-1363 ◽  
Author(s):  
Yonatan Harpaz ◽  
Matan Prasma
Keyword(s):  
Model Categories ◽  
Grothendieck Construction

scholarly journals Generalised BGP Reflection Functors via the Grothendieck Construction

2019 ◽  
Author(s):  
Tobias Dyckerhoff ◽  
Gustavo Jasso ◽  
Tashi Walde
Keyword(s):  
Grothendieck Construction ◽  
Reflection Functors

Abstract Inspired by work of Ladkani and Groth–Šťovíček, we explain how to construct generalisations of the classical reflection functors of Bernšteĭn, Gel′fand, and Ponomarev by means of the Grothendieck construction.

scholarly journals Factorizing the $$\mathbf {Top}$$–$$\mathbf {Loc}$$ adjunction through positive topologies

2021 ◽  
Author(s):  
Francesco Ciraulo ◽  
Tatsuji Kawai ◽  
Samuele Maschio
Keyword(s):  
Topological Spaces ◽  
Grothendieck Construction

AbstractWe characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

scholarly journals Network Models from Petri Nets with Catalysts

2019 ◽  
Vol 1 ◽  
pp. 4
Author(s):  
John C. Baez ◽  
John Foley ◽  
Joe Moeller
Keyword(s):  
Petri Nets ◽  
Network Model ◽  
Petri Net ◽  
Network Models ◽  
Monoidal Structure ◽  
Grothendieck Construction ◽  
Compositional Design

Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a `catalyst': an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.

2-Dimensional Categories

2021 ◽  
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Tensor Product ◽  
Category Theory ◽  
Basic Category ◽  
Double Categories ◽  
Grothendieck Construction ◽  
Monoidal Bicategories ◽  
Coherence Theorem ◽  
First Time ◽  
Yoneda Lemma

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

The Grothendieck Construction

2021 ◽  
pp. 371-438
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Detailed Proof ◽  
Grothendieck Construction ◽  
Lax Functor

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

scholarly journals The enriched Grothendieck construction

2019 ◽  
Vol 344 ◽  
pp. 234-261 ◽  
Author(s):  
Jonathan Beardsley ◽  
Liang Ze Wong
Keyword(s):  
Grothendieck Construction
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