Extension (simplicial set)#Ex∞ functor

Using the subdivision of simplicial sets, the extension of simplicial sets is defined as:Cisinski 2019, p. 81Guillou, Definition 6

: $$

\operatorname{Ex}\colon

\mathbf{sSet}\rightarrow\mathbf{sSet},

\operatorname{Ex}(Y)_n

:=\operatorname{Hom}(\operatorname{Sd}(\Delta^n),Y).

Due to the Yoneda lemma, one also has $\backslash operatorname\{Ex\}(Y)\_n$

\cong\operatorname{Hom}(\Delta^n,\operatorname{Ex}(Y))

. All connecting maps of the sets are given by precomposition with the application of the subdivision functor to all canonical inclusions $\backslash Delta^\{n-1\}\backslash hookrightarrow\backslash Delta^n$. Since the subdivision functor by definition commutes with all colimits, and for every simplicial set $X$ there is an isomorphism:Guillou, Proposition 1

: $X\backslash cong\backslash varinjlim\_\{\backslash Delta^n\backslash rightarrow\; X\}\backslash Delta^n,$

it is in fact left adjoint to the extension functor, denoted $\backslash operatorname\{Sd\}\backslash dashv\backslash operatorname\{Ex\}$. For simplicial sets $X$ and $Y$, one has:

: $$

\begin{align}

\operatorname{Hom}(\operatorname{Sd}(X),Y)

&\cong\operatorname{Hom}(\operatorname{Sd}(\varinjlim_{\Delta^n\rightarrow X}\Delta^n),Y)

\cong\operatorname{Hom}(\varinjlim_{\Delta^n\rightarrow X}\operatorname{Sd}(\Delta^n),Y)

\cong\varprojlim_{\Delta^n\rightarrow X}\operatorname{Hom}(\operatorname{Sd}(\Delta^n),Y) \\

&\cong\varprojlim_{\Delta^n\rightarrow X}\operatorname{Hom}(\Delta^n,\operatorname{Ex}(Y))

\cong\operatorname{Hom}(\varinjlim_{\Delta^n\rightarrow X}\Delta^n,\operatorname{Ex}(Y))

\cong\operatorname{Hom}(X,\operatorname{Ex}(Y)).

\end{align}

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:

: $X\backslash cong\backslash varinjlim\_\{|\backslash Delta^n|\backslash rightarrow\; X\}|\backslash Delta^n|$

for every topological space $X$. This is because the colimit is always a CW complex, for which the isomorphism does indeed hold.

The natural transformation $a\backslash colon\backslash operatorname\{Sd\}\backslash Rightarrow\backslash operatorname\{Id\}$ induces a natural transformation $b\backslash colon$

\operatorname{Id}\Rightarrow\operatorname{Ex} under the adjunction $\backslash operatorname\{Sd\}\backslash dashv\backslash operatorname\{Ex\}$. In particular there is a canonical morphism $b\_X\backslash colon$

X\rightarrow\operatorname{Ex}(X) for every simplicial set $X$.

For a simplicial set $X$, the canonical morphism $b\_X\backslash colon$

X\rightarrow\operatorname{Ex}(X) indudes an $\backslash mathbb\{N\}$-shaped cone $X\backslash rightarrow\backslash operatorname\{Ex\}(X)\backslash rightarrow\backslash operatorname\{Ex\}^2(X)\backslash rightarrow\backslash operatorname\{Ex\}^3(X)\backslash rightarrow\backslash ldots$, whose limit is denoted:Cisinski 2019, Equation (3.1.22.4)Guillou, Definition 7

: $$

\operatorname{Ex}^\infty(X)

:=\varinjlim_{n\in\mathbb{N}}\operatorname{Ex}^n(X).

Since limit and colimit are switched, there is no adjunction $\backslash operatorname\{Sd\}^\backslash infty\backslash dashv\backslash operatorname\{Ex\}^\backslash infty$ with the Sd∞ functor. But for the study of simplices, this is of no concern as any $m$-simplex $\backslash Delta^m\backslash rightarrow\backslash operatorname\{Ex\}^\backslash infty(X)$ due to the compactness of the standard $m$-simplex $\backslash Delta^m$ factors over a morphism $\backslash Delta^m\backslash rightarrow\backslash operatorname\{Ex\}^n(X)$ for a $n\backslash in\backslash mathbb\{N\}$, for which the adjunction $\backslash operatorname\{Sd\}^n\backslash dashv\backslash operatorname\{Ex\}^n$ can then be applied to get a morphism $\backslash operatorname\{Sd\}^n(\backslash Delta^m)\backslash rightarrow\; X$.

The natural transformation $b\backslash colon$

\operatorname{Id}\Rightarrow\operatorname{Ex} induces a natural transformation $\backslash beta\backslash colon$

\operatorname{Id}\Rightarrow\operatorname{Ex}^\infty. In particular there is a canonical morphism $\backslash beta\_X\backslash colon$

X\rightarrow\operatorname{Ex}^\infty(X) for every simplicial set $X$.

• For every simplicial set $X$, the canonical morphism $b\_X\backslash colon$

X\rightarrow\operatorname{Ex}(X) is a weak homotopy equivalence.Goerss & Jardine 1999, Theorem 4.6.Cisinski 2019, Proposition 3.1.21

• The extension functor $$

\operatorname{Ex}

preserves weak homotopy equivalences (which follows directly from the preceding property and their 2-of-3 property) and Kan fibrations,Goerss & Jardine 1999, Lemma 4.5. for Kan fibrations for Ex hence fibrations and trivial fibrations of the Kan–Quillen model structure. This makes the adjunction $\backslash operatorname\{Sd\}\backslash dashv\backslash operatorname\{Ex\}$ even into a Quillen adjunction $\backslash operatorname\{Sd\}\backslash colon$

\mathbf{sSet}_\mathrm{KQ}\rightleftarrows\mathbf{sSet}_\mathrm{KQ}\colon

\operatorname{Ex}.

• For every horn inclusion $\backslash Lambda\_k^n\backslash hookrightarrow\backslash operatorname\{Ex\}(X)$ with a simplicial set $X$ there exists an extension $\backslash Delta^n\backslash hookrightarrow\backslash operatorname\{Ex\}^2(X)$.Goerss & Jardine 1999, Lemma 4.7.Guillou, Lemma 1
• For every simplicial set $X$, the simplicial set $\backslash operatorname\{Ex\}^\backslash infty(X)$ is a Kan complex, hence a fibrant object of the Kan–Quillen model structure.Goerss & Jardine 1999, Theorem 4.8. on p. 188Cisinski 2019, Theorem 3.1.27Guillou, Properties of Ex∞ This follows directly from the preceding property. Furthermore the canonical morphism $\backslash beta\_X\backslash colon$

X\hookrightarrow\operatorname{Ex}^\infty(X) is a monomorphism and a weak homotopy equivalence, hence a trivial cofibration of the Kan–Quillen model structure. $\backslash operatorname\{Ex\}^\backslash infty(X)$ is therefore the fibrant replacement of $X$ in the Kan–Quillen model structure, hence the factorization of the terminal morphism $X\backslash rightarrow\backslash Delta^0$ in a trivial cofibration followed by a fibration. Furthermore, there is a restriction $$

\operatorname{Ex}^\infty\colon

\mathbf{sSet}\rightarrow\mathbf{Kan}

with the subcategory $$

\mathbf{Kan}\hookrightarrow\mathbf{sSet}

of Kan complexes.

• The infinite extension functor $$

\operatorname{Ex}^\infty

preserves all three classes of the Kan–Quillen model structure, hence Kan fibrations, monomorphisms and weak homotopy equivalences (which again follows directly from the preceding property and their 2-of-3 property).Cisinski 2019, Proposition 3.1.23.Goerss & Jardine 1999, Theorem 4.8. (3) for Kan fibrations for Ex∞

• The extension functor $$

\operatorname{Ex}

and the infinite extension functor $$

\operatorname{Ex}^\infty

both preserve the set of $0$-simplices, which follows directly from $$

\operatorname{Sd}(\Delta^0)

\cong\Delta^0

. For a simplicial set $X$, one has:Cisinski 2019, 3.8.6.

• :$$

\operatorname{Ex}(X)_0

=X_0,

• : $$

\operatorname{Ex}^\infty(X)_0

=X_0.

• The extension functor fixes the singular functor. For a topological space $X$, one has:

:: $$

\operatorname{Ex}\operatorname{Sing}(X)

\cong\operatorname{Sing}(X).

: This follows from $$

|\operatorname{Sd}(X)|

\cong|X|

for every simplicial set $X${{Cite web |last=Lurie |first=Jacob |title=Kerodon, Proposition 3.3.3.7. |url=https://kerodon.net/tag/00YB |access-date=2025-04-19 |website=kerodon.net |language=en}} by using the adjunctions $|-|\backslash dashv\backslash operatorname\{Sing\}$ and $\backslash operatorname\{Sd\}\backslash dashv\backslash operatorname\{Ex\}$. In particular, for a topological space $X$, one has:

:: $$

\operatorname{Ex}^\infty\operatorname{Sing}(X)

\cong\operatorname{Sing}(X),

: which fits the fact that the singular functor already produces a Kan complex, which can be its own fibrant replacement.

• {{cite book |last1=Goerss |first1=Paul |url=http://www.math.uwo.ca/~jardine/papers/simp-sets/ |title=Simplicial homotopy theory |last2=Jardine |first2=John Frederick |date=1999 |publisher= |isbn= 978-3-0346-0188-7|series=Modern Birkhäuser Classics |language=en |doi=10.1007/978-3-0346-0189-4|archive-url=http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/ |archive-date=8 February 1999 }}
• {{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}
• {{cite web |last=Guillou |first=Bertrand |title=Kan's Ex∞ functor |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |language=en}}

• Kan fibrant replacement at the nLab
• [https://kerodon.net/tag/00XF The Ex∞ Functor] at Kerodon
• [https://mathoverflow.net/questions/324738/why-is-kans-ex-infty-functor-useful Why is Kan's Ex∞ functor useful?] on MathOverflow

Category:Higher category theory