path space object in nLab (original) (raw)

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Model category theory

model category, model ∞ \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)(\infty,1)-categories

Model structures

for ∞\infty-groupoids

for ∞-groupoids

for equivariant ∞\infty-groupoids

for rational ∞\infty-groupoids

for rational equivariant ∞\infty-groupoids

for nn-groupoids

for ∞\infty-groups

for ∞\infty-algebras

general ∞\infty-algebras

specific ∞\infty-algebras

for stable/spectrum objects

for (∞,1)(\infty,1)-categories

for stable (∞,1)(\infty,1)-categories

for (∞,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (∞,1)(\infty,1)-sheaves / ∞\infty-stacks

Contents

Idea

A path space object in homotopy theory is an object in a homotopical category that behaves for many purposes as the topological path space does in topological homotopy theory.

Definition

Definition

(path space object)
For 𝒞\mathcal{C} a category with weak equivalences and with binary products, a path space object of/for an object XX of 𝒞\mathcal{C} is a factorization of the diagonal morphism X→(Id,Id)X×XX \stackrel{(Id, Id)}{\to} X \times X into the product as

X→sPaths X→(d 0,d 1)X×X X \xrightarrow{\;\;\; s \;\;\;} Paths_X \xrightarrow{\;\;\; (d_0, d_1) \;\;\;} X \times X

such that ss is a weak equivalence (Quillen 1967, §I.1).

Moreover (Dwyer & Spalinski 1995, §4.12):

If 𝒞\mathcal{C} in addition has the structure of a fibration category then one speaks, furthermore, of a good path space object if (d 0,d 1)(d_0,d_1) is a fibration.

If 𝒞\mathcal{C} furthermore has the structure of a model category then one speaks of a very good path space object if (d 0,d 1)(d_0,d_1) is a fibration and ss is a cofibration (hence an acyclic cofibration).

Examples

In model categories

If CC is a model category then the factorization axiom ensures that for every object X∈CX \in C there is a factorization of the diagonal

X→≃X I→(d 0,d 1)X×X X \stackrel{\simeq}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X

with the additional property that X I→X×XX^I \to X \times X is a fibration.

If XX itself is fibrant, then the projections X×X→XX \times X \to X are fibrations and moreover by 2-out-of-3 applied to the diagram

X I s↗ ↘ d i X →Id X \array{ && X^I \\ & {}^{\mathllap{s}}\nearrow && \searrow^{\mathrlap{d_i}} \\ X &&\stackrel{Id}{\to}&& X }

are themselves weak equivalences X I→≃XX^I \stackrel{\simeq}{\to} X. This is a key property that implies the factorization lemma.

If moreover the small object argument applies in the model category CC, then such factorizations, and hence path objects, may be chosen functorially: such that for each morphism X→YX \to Y the factorizations fit into a commuting diagram

X →≃ X I → X×X ↓ ↓ ↓ Y →≃ Y I → Y×Y \array{ X &\stackrel{\simeq}{\to} &X^I &\to & X \times X \\ \downarrow && \downarrow && \downarrow \\ Y & \stackrel{\simeq}{\to} & Y^I &\to & Y \times Y }

In simplicial model categories

If CC is a simplicial model category, then the powering over sSet can be used to explicitly construct functorial path objects for fibrant objects XX: define X→X I→X×XX \to X^I \to X \times X to be the powering of XX by the morphisms

Δ[0]∐Δ[0]↪d 0,d 1Δ[1]→≃Δ[0] \Delta[0] \coprod \Delta[0] \stackrel{d_0, d_1}{\hookrightarrow} \Delta[1] \stackrel{\simeq}{\to} \Delta[0]

in sSet QuillensSet_{Quillen}. Notice that the first morphism is a cofibration and the second a weak equivalence in the standard model structure on simplicial sets and that all objects are cofibrant.

Since by the axioms of an enriched model category the powering functor

(−) (−):sSet op×C→C (-)^{(-)} : sSet^{op} \times C \to C

sends cofibrations and acyclic cofibrations in the first argument to fibrations and acyclic fibrations inif the second argument is fibrant, and since this implies by the factorization lemma that it then also preserves weak equivalences between cofibrant objects, it follows that X Δ[1]X^{\Delta[1]} is indeed a path object with the extra property that also the two morphisms X Δ[1]→XX^{\Delta[1]} \to X are acyclic fibrations.

Right homotopies

Path objects are used to define a notion of right homotopy between morphisms in a category. Thus they capture aspects of higher category theory in a 11-categorical context.

Loop space objects

From a path space object may be derived loop space objects.

References

The general definition in model categories is due to:

The terminology of “good” and “very good” path space objects appears in:

Lecture notes:

Last revised on March 16, 2025 at 16:37:58. See the history of this page for a list of all contributions to it.