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, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A category with path objects or a path category is a category $\mathcal{C}$ equipped with two classes of morphisms called weak equivalences and fibrations, such that:
Fibrations are closed under composition
The pullback of a fibration along any other map exists and is again a fibration.
The pullback of an acyclic fibration along any other map is again an acyclic fibration.
Weak equivalences satisfy the two-out-of-six property: if $f:A \to B$, $g:B \to C$, $h:C \to D$ are three composable maps and both $g \circ f$ and $h \circ g$ are weak equivalences, then so are $f$, $g$, $h$ and $h \circ g \circ f$.
Isomorphisms are acyclic fibrations and every acyclic fibration has a section.
For any object $B$ there is a path space object $P B$ (not necessarily functorial in $B$).
$\mathcal{C}$ has a terminal object $1$ and every map $X \to 1$ to the terminal object is a fibration.
The syntactic category associated to a dependent type theory with identity types is a category with path objects.
Given a Quillen model category $\mathcal{C}$ in which every object is cofibrant, the full subcategory of fibrant objects in $\mathcal{C}$ is a category with path objects.
The category of topological spaces is a category with path objects where the homotopy equivalences are the weak equivalences and the Hurewicz fibrations are the the fibrations.
A finitely complete category is a category with path objects in which every morphism is a fibration and only the isomorphisms are weak equivalences.
Benno van den Berg, Ieke Moerdijk, Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories, Journal of Pure and Applied Algebra 222 10 (2018) 3137-3181 [doi:10.1016/j.jpaa.2017.11.017, arXiv:1603.02456]
Benno van den Berg, Martijn den Besten, Quadratic type checking for objective type theory (arXiv:2102.00905)
Last revised on July 25, 2023 at 15:02:32. See the history of this page for a list of all contributions to it.