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A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

The key here is the regularity of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute (that is, given some diagrams commuting in the limit, we can find $k$-small subset B for which the diagram commutes in the limit of B.

A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

The key here is the regularity of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute (that is, given some diagrams commuting in the limit, we can find $k$-small subset B for which the diagram commutes in the limit of B.

A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

The key here is the regularity of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute.

A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

The key here is the regularity of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute (that is, given some diagrams commuting in the limit, we can find $k$-small subset B for which the diagram commutes in the limit of B.

A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal see this MO question. To see it put to use in homotopy theory, take a look at Lurie's Higher topos theory A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).

The key here is the regularity of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute.