path space object


Homotopy theory

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A path space object is an internalisation of a path space.


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

XsX I(d 0,d 1)X×X X \stackrel{s}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X

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


Here C IC^I is a primitive symbol. II is not assumed to be an object and C IC^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(d 0,d 1)C×CC^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.


In model categories

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

XX 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 IX×XX^I \to X \times X is a fibration.

If XX itself is fibrant, then the projections X×XXX \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 IXX^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 XYX \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 XX IX×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×CC (-)^{(-)} : 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.

Revised on April 13, 2014 23:46:37 by Adeel Khan (