nLab function complex



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Paths and cylinders

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Basic facts




In categories of simplicial objects, like SimplicialSets, the sSet-value hom-object between two objects is often called the function complex (“simplicial complex of functions from XX to YY).

The simplicial mapping complex or simplicial hom complex of simplicial function complex of two simplicial sets is the collection of maps, left homotopies and higher homotopies between them, all itself organized into a simplicial set.

More formally, SimplicialSets is a Cartesian monoidal category and the corresponding internal hom is what is traditionally known as the simplicial mapping complex.



Suppose 𝒞\mathcal{C} is a category that admits small coproducts.

Given simplicial objects X,A𝒞 Δ opX,A\in \mathcal{C}^{\Delta^{op}}, their function complex is the simplicial set

Hom Δ(X,A)sSet Hom_{\Delta}(X,A) \;\; \in \;\; sSet

whose set of nn-simplices is the set of maps Δ nAB\Delta^n\otimes A\to B

(Hom Δ(X,A)) n=Hom 𝒞(Δ nX,A), \big( Hom_{\Delta}(X,A) \big)_n \;\; = \;\; Hom_{\mathcal{C}} \big( \Delta^n \otimes X ,\, A \big) \,,

where \otimes denotes the copowering of simplicial objects over simplicial sets given by

(SX) n= iS nX n. (S \otimes X)_n \,=\, \coprod_{i \in S_n} X_n \,.

In simplicial sets

Specializing the general definition above to 𝒞=\mathcal{C} = sSet itself, the required tensoring is just the product of simplicial sets and so


The function complex between simplicial sets X,AX, A is the cartesian internal hom given by

X,AsSet{Maps(X,A)sSet Maps(X,A) =Hom sSet(Δ[]×X,A) X, A \,\in\, sSet \;\;\;\; \vdash \;\;\;\; \left\{ \, \begin{array}{l} Maps(X,A) \;\; \in \;\; sSet \\ Maps(X,A)_\bullet \;\; = \;\; Hom_{sSet}\big( \Delta[\bullet] \times X ,\, A \big) \end{array} \right.

(e.g. Goerss & Jardine (2009), I.5)


In simplicial sets


(component formula for evaluation map on function complexes)
In SimplicialSets, the evaluation map of the function complex (from Def. ) — defined as the counit of the (X×())Maps(X,)\big(X \times (-)\big)\dashv Maps(X,-)-adjunction — is given by:

X×Maps(X,A) ev A X A A X n×Maps(X,A) nHom(X×Δ[n],A) (ev A X) n A n (x,f) f(x,ι n), \begin{array}{cc} X \times Maps(X,A) &\xrightarrow{\;\; ev^X_A \;\;}& A \\ \phantom{A} \\ X_n \times \overset{ \mathclap{ Hom(X \times \Delta[n], A) } }{ \overbrace{ Maps(X,A)_n } } &\xrightarrow{\; \big(ev^X_A\big)_n \;}& A_n \\ \big( x ,\, f \big) &\mapsto& f(x, \iota_n) \,, \end{array}


ι nid Δ[n]Hom sSet(Δ[n],Δ[n])(Δ[n]) n \iota_n \;\coloneqq\; id_{\Delta[n]} \;\in\; Hom_{sSet}\big( \Delta[n],\, \Delta[n] \big) \;\simeq\; (\Delta[n])_n

denotes the unique non-degenerate nn-cell inside Δ[n]\Delta[n].

(e.g. Goerss & Jardine (2009), p. 20)


(simplicial nerve intertwines function complexes with functor categories)

Let 𝒳,𝒜\mathcal{X}, \mathcal{A} be small strict categories and write Maps(𝒳,𝒜)Maps(\mathcal{X}, \mathcal{A}) for their functor category. Then under the simplicial nerve N:Cat smsSet N \;\colon\; Cat^{sm} \longrightarrow sSet this is isomorphic to the function complex (Def. ) between their separate simplicial nerves:

N(Maps(𝒳,𝒜))Maps(N(𝒳),N(𝒜)). N \big( Maps(\mathcal{X},\,\mathcal{A}) \big) \;\simeq\; Maps \big( N(\mathcal{X}) ,\, N(\mathcal{A}) \big) \,.

(For the proof see here at nerve.)




(simplicial mapping complex between nerves of groupoids) For N(𝒢 i)N(\mathcal{G}_i) the nerves of groupoids, their simplicial mapping complex is isomorphic to the nerve of the functor groupoid from 𝒢 1\mathcal{G}_1 to 𝒢 2\mathcal{G}_2:

[N(𝒢 1),N(𝒢 2)]N(Func(𝒢 1,𝒢 2))SimplicialSets \big[ N(\mathcal{G}_1), \, N(\mathcal{G}_2) \big] \;\; \simeq \;\; N \big( Func(\mathcal{G}_1, \, \mathcal{G}_2) \big) \;\;\; \in \; SimplicialSets

Nerves of inertia groupoids


(evaluation map on function complexes which are nerves of inertia groupoids)

Consider a discrete group GG. We write BG\mathbf{B}G for its delooping groupoid, here specifically for its strict version BG(G*)\mathbf{B}G \,\coloneqq\,\big(G \rightrightarrows \ast \big). The nerve of this groupoid is the canonical model for the simplicial classifying space: NBG=W¯G. N \mathbf{B}G \,=\, \overline{W}G \,. The functor groupoid

ΛBGMaps(B,BG) \Lambda \mathbf{B}G \;\coloneqq\; Maps\big( \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big)

is also known as the inertia groupoid of BG\mathbf{B}G.

The simplicial nerve of this functor groupoid is the following function complex of simplicial sets (using Prop. ):

NΛBGMaps(NB,NBG)Maps(S,NBG). N \Lambda \mathbf{B}G \;\coloneqq\; Maps\big( N \mathbf{B}\mathbb{Z} ,\, N \mathbf{B}G \big) \;\simeq\; Maps\big( S ,\, N \mathbf{B}G \big) \,.

Here in the second step we introduced the simplicial set

(1)SΔ[1]/Δ[1]sSet, S \;\coloneqq\; \Delta[1]/\partial \Delta[1] \;\;\; \in \; sSet \,,

which is the minimal model for B\mathbf{B}\mathbb{Z}, whose unique non-degenerate 1-cell we will denote by

[0,1]Δ[1] 1S 1 \ell \;\coloneqq\; [0,1] \in \Delta[1]_1 \twoheadrightarrow S_1

We now describe in detail the evaluation map on the simplicial function complex Maps(S,NBG). Maps\big( S ,\, N \mathbf{B}G \big) \,.

To that end, we denote by

(2)(γ;g n1,g n2,,g 0)Hom(S×Δ[n],W¯G)(NMap(B,BG)) n \big( \gamma ;\, g_{n-1} ,\, g_{n-2} ,\, \cdots ,\, g_{0} \big) \;\; \in \;\; \mathrm{Hom} \big( S \times \Delta[n] ,\, \overline{W}G \big) \;\simeq\; \Big( N \mathrm{Map} \big( { \mathbf{B}\mathbb{Z} } ,\, { \mathbf{B}G } \big) \Big)_{n}

the nn-cell in the nerve of the inertia groupoid that corresponds to the sequence of natural transformation which start at the functor

γGHom Grp(,G)Hom(B,BG) \gamma \,\in\, G \,\simeq\, \mathrm{Hom}_{\mathrm{Grp}}(\mathbb{Z}, G) \,\simeq\, \mathrm{Hom}\big( \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big)

and successively have components g nGg_{n-\bullet} \in G.

Throughout we are writing “HomHom” for hom-sets and “MapMap” for hom-objects, i.e. for internal homs and we keep tacitly going back and forth through the bijections in (2).

If we abbreviate (this follows conventions familiar in discussion of transgression in group cohomology)

Ad k(γ) Ad (g n1g nk)(γ) (g n1g nk) 1γ(g n1g nk), \begin{aligned} Ad_{k}(\gamma) & \;\coloneqq\; Ad_{(g_{n-1}\cdots g_{n-k})}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_{n-k})^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_{n-k}) \,, \end{aligned}

then this corresponds to a sequence of composable morphisms in the inertia groupoid of this form:

γg n1Ad 1(γ)g n2g 0Ad n(γ)ΛBG \gamma \xrightarrow{ g_{n-1} } Ad_{1}(\gamma) \xrightarrow{ g_{n-2} } \cdots \xrightarrow{\; g_0 \;} Ad_{n}(\gamma) \;\;\;\;\; \in \;\; \Lambda B G

The simplicial map (2) maps the non-degenerate (n+1)(n+1)-cells in the product of simplicial sets (see the discussion there) of SS (1) with the simplicial n n -simplex as follows:

This is not exactly what we need unwinding the evaluation map, but it is close: the image of (2) under the kkth degeneracy map evidently gives the following mapping:

This is the type of mapping that appears in the component formula for the evaluation map on function complexes (from Prop. ). Hence we find that this evaluation map is given on non-degenerate (n+1)(n+1)-simplices in the product (see again there) as follows:

The expressions on the right happen to be those that appear in the formula for transgression in group cohomology, see there.


The original definition of a function complex in the generality stated above:

  • Daniel M. Kan, On c.s.s. categories, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

Textbook account in simplicial homotopy theory:

Introduction with an eye towards quasicategories:

Last revised on December 5, 2022 at 15:06:52. See the history of this page for a list of all contributions to it.