### Abstract

Language | English |
---|---|

Pages | 189-271 |

Number of pages | 83 |

Journal | Algebraic and geometric topology |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- ∞–categories
- modules
- profunctors
- virtual equipment
- pointwise Kan extension
- Virtual equipment
- Modules
- Profunctors
- Pointwise Kan extension

### Cite this

*Algebraic and geometric topology*,

*17*(1), 189-271. https://doi.org/10.2140/agt.2017.17.189

}

*Algebraic and geometric topology*, vol. 17, no. 1, pp. 189-271. https://doi.org/10.2140/agt.2017.17.189

**Kan extensions and the calculus of modules for ∞–categories.** / Riehl, Emily; Verity, Dominic.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Kan extensions and the calculus of modules for ∞–categories

AU - Riehl,Emily

AU - Verity,Dominic

PY - 2017

Y1 - 2017

N2 - Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.

AB - Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.

KW - ∞–categories

KW - modules

KW - profunctors

KW - virtual equipment

KW - pointwise Kan extension

KW - Virtual equipment

KW - Modules

KW - Profunctors

KW - Pointwise Kan extension

UR - http://purl.org/au-research/grants/arc/DP130101969

UR - http://www.scopus.com/inward/record.url?scp=85011875134&partnerID=8YFLogxK

U2 - 10.2140/agt.2017.17.189

DO - 10.2140/agt.2017.17.189

M3 - Article

VL - 17

SP - 189

EP - 271

JO - Algebraic and geometric topology

T2 - Algebraic and geometric topology

JF - Algebraic and geometric topology

SN - 1472-2739

IS - 1

ER -