nLab (n+1,1)-category of n-truncated objects



Category theory

2-ategory theory

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




Given any object XX in any (k,1)(k,1)-category CC for a natural number kk, the nn-truncated objects of XX for n+1<kn+1 \lt k form an (n+1,1)(n+1,1)-category, an (n+1)(n+1)-truncated (k,1)(k,1)-category, (though in general it may be a large one, cf. well-powered category). This is called the (n+1,1)(n+1,1)-category of nn-truncated objects of XX, or the nn-truncated object (n+1,1)(n+1,1)-category of XX.


If CC is finitely complete, then the nn-truncated objects form a finitely complete (n+1,1)(n+1,1)-category, so we may speak of the finitely complete (n+1,1)(n+1,1)-category of nn-truncated objects.

In any coherent (k,1)(k,1)-category, then the nn-truncated objects form a coherent (n+1,1)(n+1,1)-category, so we may speak of the coherent (n+1,1)(n+1,1)-category of nn-truncated objects.

In any (k,1)(k,1)-topos, the nn-truncated objects of XX form a (n+1,1)(n+1,1)-topos, so we may speak of the (n+1,1)(n+1,1)-topos of nn-truncated objects.

The reader can probably think of other variations on this theme.

If f:XYf : X \to Y is a morphism that has pullbacks along nn-truncated morphisms, then pullback along ff induces a (n+1,1)(n+1,1)-category morphism f *:Trunc n(Y)Trunc n(X)f^* : Trunc_n(Y) \to Trunc_n(X). This is functorial in the sense that if g:YZg : Y \to Z also has this property, then there is an inhabited equivalence n+1n+1-groupoid f *g *(gf) *f^* \circ g^* \cong (g \circ f)^*.

If CC has pullbacks of nn-truncated morphisms, Trunc nTrunc_n is often used to denote the contravariant functor C op(n+1,1)CatC^{op} \to (n+1,1)Cat whose action on morphisms is Trunc n(f)=f *Trunc_n(f) = f^*.


  • If one opts for the alternative definition that nn-truncated objects are nn-truncated morphisms into the object (not equivalence large n+1n+1-groupoids thereof), then one gets a (n+1,1)(n+1,1)-precategory of nn-truncated objects instead. In any case, the (n+1,1)(n+1,1)-category of nn-truncated objects Trunc n(X)Trunc_n(X) in our sense is the (n+1,1)(n+1,1)-categorical reflection of the (n+1,1)(n+1,1)-precategory TruncMor n(X)TruncMor_n(X) of nn-truncated objects in the alternative sense, and of course the reflection Rezk completion map TruncMor n(X)Trunc n(X)TruncMor_n(X) \to Trunc_n(X) is an equivalence.

  • poset of subobjects

  • n-truncated object classifier?

Last revised on June 12, 2021 at 14:34:11. See the history of this page for a list of all contributions to it.