nLab
path space object

Idea

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

Definition

In $C$ a category with weak equivalences and with products a path space object of an object $X$ is a factorization of the diagonal morphism $X \stackrel{(Id, Id)}{\to} X \times X$ into the product as

$$X \stackrel{s}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X$$

such that $s$ is a weak equivalence. (This also makes sense even if the product $X \times X$ doesn’t exist.) We interpret a (generalised) element of $X^I$ as a path in $X$.

Notice

Here $C^I$ is a primitive symbol. $I$ is not assumed to be an object and $C^I$ is not assumed to be an internal hom. This is standard but somewhat abusive notation. It is supposed to remind us of the “nice” situation where the path object is co-represented by an interval object.

If the category in question also has a notion of fibrations, such as in a category of fibrant objects or in a model category, the morphism $C^I \stackrel{(d_0, d_1)}{\to} C \times C$ in the definition of a path object is required to be a fibration.

Path space objects are in particular guaranteed to exist in any model category.

Examples

In model categories

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

$$X \stackrel{\simeq}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X$$

with the additional property that $X^I \to X \times X$ is a fibration.

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

$$\array{ && X^I \\ & {}^{\mathllap{s}}\nearrow && \searrow^{\mathrlap{d_i}} \\ X &&\stackrel{Id}{\to}&& X }$$

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

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

$$\array{ X &\stackrel{\simeq}{\to} &X^I &\to & X \times X \\ \downarrow && \downarrow && \downarrow \\ Y & \stackrel{\simeq}{\to} & Y^I &\to & Y \times Y }$$

In simplicial model categories

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

$$\Delta[0] \coprod \Delta[0] \stackrel{d_0, d_1}{\hookrightarrow} \Delta[1] \stackrel{\simeq}{\to} \Delta[0]$$

in $sSet_{Quillen}$. 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

$$(-)^{(-)} : sSet^{op} \times C \to C$$

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 $X^{\Delta[1]}$ is indeed a path object with the extra property that also the two morphisms $X^{\Delta[1]} \to X$ are acyclic fibrations.

Related notions

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 $1$-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:

• Daniel Quillen, Section I.1, Def. 4, p. 9 (15 of 165) in: Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967 (doi:10.1007/BFb0097438)

Lecture notes:

• Introduction to Homotopy Theory, around this Def.