2-Category theory

Limits and colimits



An inserter is a particular kind of 2-limit in a 2-category, which universally inserts a 2-morphism between a pair of parallel 1-morphisms.


Let f,g:ABf,g\colon A\rightrightarrows B be a pair of parallel 1-morphisms in a 2-category. The inserter of ff and gg is a universal object VV equipped with a morphism v:VAv\colon V\to A and a 2-morphism α:fvgv\alpha\colon f v \to g v.

More precisely, universality means that for any object XX, the induced functor

Hom(X,V)Ins(Hom(X,f),Hom(X,g))Hom(X,V) \to Ins(Hom(X,f),Hom(X,g))

is an equivalence, where Ins(Hom(X,f),Hom(X,g))Ins(Hom(X,f),Hom(X,g)) denotes the category whose objects are pairs (u,β)(u,\beta) where u:XAu\colon X\to A is a morphism and β:fugu\beta\colon f u \to g u is a 2-morphism. If this functor is an isomorphism of categories, then we say that VvAV\xrightarrow{v} A is a strict inserter.

Inserters and strict inserters can be described as a certain sort of weighted 2-limit, where the diagram shape is the walking parallel pair P=()P = (\cdot \rightrightarrows\cdot) and the weight PCatP\to Cat is the diagram

1I1 \;\rightrightarrows\; I

where 11 is the terminal category and II is the interval category. Note that inserters are not equivalent to any sort of conical 2-limit.

An inserter in K opK^{op} (see opposite 2-category) is called a coinserter in KK.

Example: the strict 2-category of categories

In the strict 2-category of categories, inserters can be concretely described as follows.

The input data is two categories, AA and BB, and two functors, F,G:ABF,G\colon A\to B. The objects of the inserter are pairs (X,b)(X,b), where XAX\in A and b:F(A)G(A)b\colon F(A)\to G(A). Morphisms (X,b)(X,b)(X,b)\to(X',b') are morphisms f:XXf\colon X\to X' such that bF(f)=G(f)bb'\circ F(f)=G(f)\circ b. The functor from the inserter to AA discards the data of bb.



  • Let CC denote a category and F:CCF : C \rightarrow C denote a functor. Then the notion of an algebra for an endofunctor of FF corresponds to the inserter of FF and id C\mathrm{id}_C, and the notion of a coalgebra for an endofunctor of FF corresponds to the inserter of id C\mathrm{id}_C and FF.


  • G.J. Bird, G.M. Kelly, A.J. Power, R.H. Street, Flexible limits for 2-categories, J. Pure and Applied Algebra 61:1, 1-27, 1989, doi:10.1016/0022-4049(89)90065-0
  • Fernando Lucatelli Nunes, Freely generated nn-categories, coinserters and presentations of low dimensional categories, arxiv/1704.04474

Last revised on March 2, 2021 at 06:25:06. See the history of this page for a list of all contributions to it.