## Algebraic & Geometric Topology

### Model structures on the category of small double categories

#### Abstract

In this paper we obtain several model structures on $DblCat$, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in $Cat$ and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, $DblCat$ inherits a model structure as a category of algebras over a $2$–monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 1855-1959.

Dates
Revised: 27 August 2008
Accepted: 27 August 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796920

Digital Object Identifier
doi:10.2140/agt.2008.8.1855

Mathematical Reviews number (MathSciNet)
MR2449004

Zentralblatt MATH identifier
1159.18302

#### Citation

Fiore, Thomas M; Paoli, Simona; Pronk, Dorette. Model structures on the category of small double categories. Algebr. Geom. Topol. 8 (2008), no. 4, 1855--1959. doi:10.2140/agt.2008.8.1855. https://projecteuclid.org/euclid.agt/1513796920

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