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

Contents

Context

Category theory

2-ategory theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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.

Properties

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^*.

Examples

  • 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 10:34:11. See the history of this page for a list of all contributions to it.