Colimits of internal categories

Calum Hughes; Adrian Miranda

doi:10.36045/j.bbms.251105

Colimits of internal categories

Calum Hughes
, Adrian Miranda

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

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Abstract

We show that for an extensive 1-category E with pullback-stable coequalisers
admitting free internal categories over internal graphs, the 2-category
Cat(E) of internal categories, functors and natural transformations has finite
2-colimits. In addition, Cat(E) is extensive and codescent coequalisers
are stable under pullback along discrete Conduch´e fibrations. Moreover, we
give converse results to this.

Original language	English
Article number	3
Pages (from-to)	351-385
Number of pages	35
Journal	Bulletin of the Belgian Mathematical Society
Volume	33
Issue number	3
DOIs	https://doi.org/10.36045/j.bbms.251105
Publication status	Published - 20 Jul 2026

Keywords

internal categories
colimits
coequalisers
2-catergory theory
2-categories
elementary topos

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10.36045/j.bbms.251105

Coequalisers_in_CatEAccepted author manuscript, 490 KBLicence: CC BY

Cite this

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title = "Colimits of internal categories",

abstract = "We show that for an extensive 1-category E with pullback-stable coequalisers admitting free internal categories over internal graphs, the 2-category Cat(E) of internal categories, functors and natural transformations has finite 2-colimits. In addition, Cat(E) is extensive and codescent coequalisers are stable under pullback along discrete Conduch´e fibrations. Moreover, we give converse results to this.",

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AU - Hughes, Calum

AU - Miranda, Adrian

PY - 2026/7/20

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N2 - We show that for an extensive 1-category E with pullback-stable coequalisers admitting free internal categories over internal graphs, the 2-category Cat(E) of internal categories, functors and natural transformations has finite 2-colimits. In addition, Cat(E) is extensive and codescent coequalisers are stable under pullback along discrete Conduch´e fibrations. Moreover, we give converse results to this.

AB - We show that for an extensive 1-category E with pullback-stable coequalisers admitting free internal categories over internal graphs, the 2-category Cat(E) of internal categories, functors and natural transformations has finite 2-colimits. In addition, Cat(E) is extensive and codescent coequalisers are stable under pullback along discrete Conduch´e fibrations. Moreover, we give converse results to this.

KW - internal categories

KW - colimits

KW - coequalisers

KW - 2-catergory theory

KW - 2-categories

KW - elementary topos

U2 - 10.36045/j.bbms.251105

DO - 10.36045/j.bbms.251105

M3 - Article

SN - 1370-1444

VL - 33

SP - 351

EP - 385

JO - Bulletin of the Belgian Mathematical Society

JF - Bulletin of the Belgian Mathematical Society

IS - 3

M1 - 3

ER -

Colimits of internal categories

Abstract

Keywords

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