Paths and cylinders
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
A path space object is an internalisation of a path space.
In a category with weak equivalences and with products a path space object of an object is a factorization of the diagonal morphism into the product as
such that is a weak equivalence. (This also makes sense even if the product doesn’t exist.) We interpret a (generalised) element of as a path in .
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 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 is a model category then the factorization axiom ensures that for every object there is a factorization of the diagonal
with the additional property that is a fibration.
If itself is fibrant, then the projections are fibrations and moreover by 2-out-of-3 applied to the diagram
are themselves weak equivalences . This is a key property that implies the factorization lemma.
If moreover the small object argument applies in the model catgeory , then such factorizations, and hence path objects, may be chosen fonctorially: such that for each morphism the factorizations fit into a commuting diagram
In simplicial model categories
If is a simplicial model category, then the powering over sSet can be used to explicitly construct functorial path objects for fibrant objcts : define to be the powering of by the morphisms
in . 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
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 is indeed a path object with the extra property that also the two morphisms are acyclic fibratios.
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 -categorical context.
Loop space objects
From a path space object may be derived loop space objects.