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awesome-double-categories

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A list of works and resources about double category theory, with a particular focus on applications.

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# Awesome Double Categories
A list of works and resources about double category theory, with a particular focus on applications.
(If you'd like to add more, edit [this nLab page](https://ncatlab.org/nlab/show/applications+of+double+category+theory))

## Introductions & references
* The Catsters, [_Double categories_](https://www.youtube.com/watch?v=kiCZiSA2W3Q) (short video)
* Grandis, [_Higher dimensional categories: from double to multiple categories_](https://doi.org/10.1142/11406), 2019,
* Johnson, Yau, [_2-Dimensional Categories_](http://arxiv.org/abs/2002.06055) (Section 12.3), 2020
* nLab, [_double category_](https://ncatlab.org/nlab/show/double+category)

## Theory
### General
* Grandis, Parè, [_Limits in double categories_](http://www.numdam.org/article/CTGDC_1999__40_3_162_0.pdf), 1999
* Grandis, Parè, [_Adjoints for double categories_](http://www.numdam.org/article/CTGDC_2004__45_3_193_0.pdf),2004
* Myers, [_String diagrams for double categories and (virtual) equipments_](https://arxiv.org/abs/1612.02762), 2016

### Equipments
* Shulman, [_Equipments_](https://golem.ph.utexas.edu/category/2009/11/equipments.html), 2009
* Shulman, [_Framed bicategories and monoidal fibrations_](https://arxiv.org/abs/0706.1286), 2009

### Grothendieck constructions
* Patterson, [_Grothendieck construction for double categories_](https://topos.site/blog/2022/05/grothendieck-construction-for-double-categories/), 2022
* Cruttwell, Lambert, Pronk, Szyld, [_Double Fibrations_](https://arxiv.org/abs/2205.15240), 2022

## Applications
### Lenses & optics
* Clarke, [_The double category of lenses_](https://bryceclarke.github.io/The_Double_Category_Of_Lenses_Phd_Thesis.pdf), 2022
* Capucci, [_Seeing double through dependent optics_](https://arxiv.org/abs/2204.10708), 2022

### Open systems theory
* Courser, [_Open systems: a double categorical perspective_](https://arxiv.org/abs/2008.02394), 2020
* Patterson, [_Decorated cospans via the double Grothendieck construction_](https://topos.site/blog/2022/05/decorated-cospans-via-the-grothendieck-construction/), 2022
* Baez, Courser, Vasilakopoulou, [_Structured versus Decorated Cospans_](https://compositionality-journal.org/papers/compositionality-4-3/pdf), 2022
* Myers, [_Categorical systems theory_](http://davidjaz.com/Papers/DynamicalBook.pdf), 2021

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