nLab category with path objects

This is about the notion of path category in model category theory and homotopy theory. For the notion of path category in ordinary category theory, see path category.


Model category theory

model category, model \infty -category



Universal constructions


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

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



Paths and cylinders

Homotopy groups

Basic facts




A category with path objects or a path category is a category 𝒞\mathcal{C} equipped with two classes of morphisms called weak equivalences and fibrations, such that:

  1. Fibrations are closed under composition

  2. The pullback of a fibration along any other map exists and is again a fibration.

  3. The pullback of an acyclic fibration along any other map is again an acyclic fibration.

  4. Weak equivalences satisfy the two-out-of-six property: if f:ABf:A \to B, g:BCg:B \to C, h:CDh:C \to D are three composable maps and both gfg \circ f and hgh \circ g are weak equivalences, then so are ff, gg, hh and hgfh \circ g \circ f.

  5. Isomorphisms are acyclic fibrations and every acyclic fibration has a section.

  6. For any object BB there is a path space object PBP B (not necessarily functorial in BB).

  7. 𝒞\mathcal{C} has a terminal object 11 and every map X1X \to 1 to the terminal object is a fibration.


See also


Last revised on July 25, 2023 at 15:02:32. See the history of this page for a list of all contributions to it.