Filtered colimits of spaces and based loops

Assume the Axiom of Choice and fixed compatible Grothendieck universes containing $\omega$. A filtered category here is a nonempty small category in which every finite diagram admits a cocone. Filtered colimits of spaces commute with finite homotopy limits. Consequently, for a filtered diagram of pointed spaces $(T_a,t_a)$, ${\mathrm{colim}}_a{\mathrm{\Omega }}_{t_a}{T}_a\simeq {\mathrm{\Omega }}_t{\mathrm{colim}}_a{T}_a\mathrm{.}$ The ordinary filtered colimit of a diagram of simplicial sets represents its colimit as spaces; equivalently, the canonical map from its homotopy colimit to its ordinary colimit is a weak equivalence. One may compute this with functorial Kan replacements.

A simplicial set is a contravariant functor from the simplex category to sets, with degreewise ordinary colimits. A Kan complex is a simplicial set in which every horn extends to a simplex. Spaces mean homotopy types of simplicial sets, using the homotopy-coherent Kan-complex model $\mathcal{S}$; homotopy limits and colimits mean limits and colimits in this model. A simplicial map is a weak equivalence when its realization induces a bijection on path components and isomorphisms on all based homotopy groups. For a pointed space, ${\mathrm{\Omega }}_{t}T=\ast {\times }_T^h\ast$ uses the two arrows selecting its basepoint. The point $t$ on the colimit is induced by the diagram of basepoints.

Write $\mathrm{\mid }T\mathrm{\mid }$ for the geometric realization obtained by gluing topological simplices according to the face and degeneracy maps of $T$. The singular simplicial set $\mathrm{Sing}V$ has as its $n$-simplices the continuous maps from the standard topological $n$-simplex to $V$, with faces and degeneracies given by precomposition. Realization and singular complex are equivalent homotopy models: the maps $T\to \mathrm{Sing}\mathrm{\mid }T\mathrm{\mid }$ and $\mathrm{\mid }\mathrm{Sing}V\mathrm{\mid }\to V$ are weak equivalences, for simplicial sets $T$ and compactly generated weak Hausdorff spaces $V$. These conventions include the nerve realizations used for Quillen K-spaces; a category's nerve has composable strings of $n$ arrows as its $n$-simplices. This contextual application imposes no further K-theory hypothesis.

The Axiom of Choice

Grothendieck universe closure convention

Simplicial sets, homotopies and trivial Kan fibrations

Simplicial horns and Kan fibrations

Weak homotopy equivalence

Compactly generated conventions for based homotopy

Singular simplices and singular chain groups with coefficients

Import precisely the finite-limit theorem of reference 0, the realization comparison of reference 1, the two filtered simplicial/Kan comparisons of reference 2, and the functorial replacement of reference 3, under [A1]. Their hypotheses apply to the small filtered diagrams and homotopy models specified here. The imported model structures, cofinality and localization arguments are not reconstructed. The proof below gives the local loop application, transports the Kan-colimit model to arbitrary simplicial sets, and checks the singular-chain identification from its definition. No continuity of motivic coefficients is asserted.