In higher category theory in mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty and strong applications in homotopical algebra. Among the most well-known is its application in the construction of Kan complexes from arbitrary simplicial sets, which often enables without loss of generality to take the former for proofs about the latter. It is furthermore very well compatible with the Kan–Quillen model structure and can for example be used to explicitly state its factorizations or to search for weak homotopy equivalences.
Definition.
Using the subdivision of simplicial sets, the extension of simplicial sets is defined as:
 formula_1
Due to the Yoneda lemma, one also has formula_2. All connecting maps of the sets are given by precomposition with the application of the subdivision functor to all canonical inclusions formula_3. Since the subdivision functor by definition commutes with all colimits, and for every simplicial set formula_4 there is an isomorphism:
 formula_5
it is in fact left adjoint to the extension functor, denoted formula_6. For simplicial sets formula_4 and formula_8, one has:
 formula_9
It is therefore possible to also simply define the extension functor as the right adjoint to the subdivision functor. Both of their construction as extension by colimits and definition is similar to that of the adjunction between geometric realization and the singular functor, with an important difference being that there is no isomorphism:
 formula_10
for every topological space formula_4. This is because the colimit is always a CW complex, for which the isomorphism does indeed hold.
The natural transformation formula_12 induces a natural transformation formula_13 under the adjunction formula_6. In particular there is a canonical morphism formula_15 for every simplicial set formula_4.
Ex∞ functor.
For a simplicial set formula_4, the canonical morphism formula_15 indudes an formula_19-shaped cone formula_20, whose limit is denoted:
 formula_21
Since limit and colimit are switched, there is no adjunction formula_22 with the Sd∞ functor. But for the study of simplices, this is of no concern as any formula_23-simplex formula_24 due to the compactness of the standard formula_23-simplex formula_26 factors over a morphism formula_27 for a formula_28, for which the adjunction formula_29 can then be applied to get a morphism formula_30.
The natural transformation formula_13 induces a natural transformation formula_32. In particular there is a canonical morphism formula_33 for every simplicial set formula_4.
 formula_60
 This follows from formula_61 for every simplicial set formula_4 by using the adjunctions formula_63 and formula_6. In particular, for a topological space formula_4, one has:
 formula_66
 which fits the fact that the singular functor already produces a Kan complex, which can be its own fibrant replacement.