nLab simplicial object



Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A simplicial object XX in a category CC is a simplicial set internal to CC: a collection {X n} n\{X_n\}_{n \in \mathbb{N}} of objects in CC that behave as if X nX_n were an object of nn-dimensional simplices internal to CC equipped with maps between these spaces that assign faces and degenerate simplices.

For instance, and there is a longer list further down this page, a simplicial object in GrpsGrps is a collection {G n} n\{G_n\}_{n\in \mathbb{N}} of groups, together with face and degeneracy homomorphisms between them. This is just a simplicial group. We equally well have other important instances of the same idea, when we replace GrpsGrps by other categories, or higher categories.


A simplicial object in a category CC is a functor Δ opC\Delta^{op} \to C, where Δ\Delta is the simplicial indexing category.

More generally, a simplicial object in an (∞,1)-category is an (∞,1)-functor Δ opC\Delta^{op} \to C.

A cosimplicial object in CC is similarly a functor out of the opposite category, ΔC\Delta \to C.

Accordingly, simplicial and cosimplicial objects in CC themselves form a category in an obvious way, namely the functor category [Δ op,C][\Delta^{op},C] and [Δ,C][\Delta,C], respectively.


A simplicial object XX in CC is often specified by the objects, X nX_n, which are the images under XX, of the objects [n][n] of Δ\Delta, together with a description of the face and degeneracy morphisms, d id_i and s js_j, which must satisfy the simplicial identities.


Category of simplicial objects

For DD a category, we write D Δ opD^{\Delta^{op}} for the functor category from Δ op\Delta^{op} to DD: its category of simplicial objects.

Simplicial enrichment


Let DD be a category with all limits and colimits. This implies that it is tensored over Set

:D×SetD. \cdot : D \times Set \to D \,.

This induces a functor

Δ op:D Δ op×sSetD Δ op \cdot^{\Delta^{op}} : D^{\Delta^{op}} \times sSet \to D^{\Delta^{op}}

which we shall also write just “\cdot”.

For X,YD Δ opX,Y \in D^{\Delta^{op}} write

D Δ op(X,Y):=Hom D Δ op(XΔ[],Y)sSet D^{\Delta^{op}}(X,Y) := Hom_{D^{\Delta^{op}}}(X \cdot \Delta[\bullet], Y) \in sSet

and for X,Y,ZD Δ opX,Y,Z \in D^{\Delta^{op}} let

D Δ op(X,Y)×D Δ op(Y,Z)D Δ op(X,Z) D^{\Delta^{op}}(X,Y) \times D^{\Delta^{op}}(Y,Z) \to D^{\Delta^{op}}(X,Z)

be given in degree nn by

(XΔ[n]Y,YΔ[n]Z)(XΔ[n]XΔ[n]×Δ[n]YΔ[n]Z). (X \cdot \Delta[n] \to Y, Y \cdot \Delta[n] \to Z) \mapsto ( X \cdot \Delta[n] \to X \cdot \Delta[n]\times \Delta[n] \to Y \cdot \Delta[n] \to Z) \,.

With the above definitions D Δ opD^{\Delta^{op}} becomes an sSet-enriched category which is both tensored as well as cotensored over sSetsSet.


We may regard the category of cosimplicial objects D ΔD^{\Delta} as an sSetsSet-enriched category using the above enrichment by identifying

D Δ(D op Δ op) op. D^{\Delta} \simeq ({D^{op}}^{\Delta^{op}})^{op} \,.

Geometric realization

If DD is already a simplicially enriched category in its own right, with powers and copowers, we can define the geometric realization of a simplicial object XD Δ opX\in D^{\Delta^{op}} as a coend:

|X|= [n]ΔΔ[n]X n |X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n

where \odot denotes the copower for the simplicial enrichment of DD. This is left adjoint to the “total singular object” functor DD Δ opD \to D^{\Delta^{op}} sending YY to the simplicial object nY Δ[n]n\mapsto Y^{\Delta[n]}, the power for the simplicial enrichment.

Perhaps surprisingly, this adjunction is even a simplicially enriched adjunction when D Δ opD^{\Delta^{op}} has its simplicial structure from Definition , even though the latter makes no reference to the given simplicial enrichment of DD. A proof can be found in RSS01, Proposition 5.4.


The original definition of simplicial objects, maps between them, and homotopies of such maps is due to Daniel M. Kan:

  • Daniel M. Kan, On the homotopy relation for c.s.s. maps, Boletín de la Sociedad Matemática Mexicana 2 (1957), 75–81. PDF.

An early discussion of simplicial objects is in:

  • Alexander Grothendieck, p. 108 (11 of 21) in: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English translation: web version).

See also:

Last revised on April 18, 2024 at 14:32:00. See the history of this page for a list of all contributions to it.