nLab path space object

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

$$X \xrightarrow{\;\;\; s \;\;\;} Paths_X \xrightarrow{\;\;\; (d_0, d_1) \;\;\;} X \times X$$

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

Moreover (Dwyer & Spalinski 1995, §4.12):

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

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

Remark

In Def. one interprets

1. a (generalised) element of $X^I$ as a path in $X$;

2. $d_0, d_1$ as the maps that send a path to its start- or endpoint, respectivelyl

3. $s$ as the map that sends a point to the path constant on that point.

Remark

(in model categories) In any model category, the factorization axioms applied to the diagonal maps immediately imply that every object has a path space object, and in fact a “very good” one. (See below.)

The very good path space objects in locally cartesian closed model categories serve as categorical semantics for the identity types in dependent type theory (homotopy type theory).

Remark

In the presence of a (good, very good) interval object

$$\ast \sqcup \ast \to I \to \ast \,,$$

the exponential objects of the form $X^I$ are (good, very good) path space objects (at least for the evident corresponding definition of “interval object”).

In particular, in a convenient category of topological spaces, with $I = [0,1]$ the standard closed interval, the mapping space $X^{[0,1]}$ is the standard path space and is a path object in the general sense of Def. .

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, § 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)

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

• William Dwyer, Jan Spalinski, §4.12 in: Homotopy theories and model categories (pdf)

  in: I. M. James, Handbook of Algebraic Topology, North Holland 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)

Lecture notes:

• Introduction to Homotopy Theory, around this Def.