nLab simplicial object

Idea

A simplicial object $X$ in a category $C$ is a simplicial set internal to $C$: a collection $\{X_n\}_{n \in \mathbb{N}}$ of objects in $C$ that behave as if $X_n$ were an object of $n$-dimensional simplices internal to $C$ 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 $Grps$ is a collection $\{G_n\}_{n\in \mathbb{N}}$ 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 $Grps$ by other categories, or higher categories.

Definition

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

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

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

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

Remark

A simplicial object $X$ in $C$ is often specified by the objects, $X_n$, which are the images under $X$, of the objects $[n]$ of $\Delta$, together with a description of the face and degeneracy morphisms, $d_i$ and $s_j$, which must satisfy the simplicial identities.

Examples

• A simplicial object in Sets is a simplicial set.

• A simplicial object in Presheaves is a simplicial presheaf.

• A simplicial object in TopologicalSpaces is a simplicial topological space.

• A simplicial object in Manifolds is a simplicial manifold.

• A simplicial object in Groups is a simplicial group.

• A simplicial object in AbelianGroups is a simplicial abelian group.

• A simplicial object in TopologicalGroups is a simplicial topological group.

• A simplicial object in Lie algebras is a simplicial Lie algebra.

• A simplicial object in Rings is a simplicial ring.

• A cosimplicial object in the category of rings (algebras) is a cosimplicial ring (cosimplicial algebra).

• A simplicial object in a category of simplicial objects is a bisimplicial object.

• A cosimplicial object in sSet is a cosimplicial simplicial set (equivalently a simplicial object in cosimplicial sets).

• The bar construction produces a simplicial object from a monad and an algebra over that monad.

Category of simplicial objects

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

Simplicial enrichment

Definition

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

$$\cdot : D \times Set \to D \,.$$

This induces a functor

$$\cdot^{\Delta^{op}} : D^{\Delta^{op}} \times sSet \to D^{\Delta^{op}}$$

which we shall also write just “$\cdot$”.

For $X,Y \in D^{\Delta^{op}}$ write

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

and for $X,Y,Z \in D^{\Delta^{op}}$ let

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

be given in degree $n$ by

$$(X \cdot \Delta[n] \to Y, Y \cdot \Delta[n] \to Z) \mapsto ( X \cdot \Delta[n] \to X \cdot \Delta[n]\times \Delta[n] \to Y \cdot \Delta[n] \to Z) \,.$$

Proposition

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

Definition

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

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

Geometric realization

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

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

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

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

Related concepts

• Reedy model structure

• simplex, simplex category

• simplicial object

  • simplicial set

  • simplicial diagram

  • simplicial object in an (∞,1)-category

• semi-simplicial object

  • semisimplicial set

• globular set, cubical set

• nerve, nerve and realization

• Segal condition

References

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:

• Peter May, Simplicial objects in algebraic topology, University of Chicago Press (1967) [ISBN:9780226511818, djvu, pdf]

• Dai Tamaki, Akira Kono, Appendix A.1 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Charles Rezk, Stefan Schwede, and Brooke Shipley, Simplicial structures on model categories and functors, arxiv

• Michael Barr, John Kennison, Robert Raphael, Contractible simplicial objects, Comm. Math. Univ. Carol. 60 4 (2019) 473–495 [pdf eudml:295068]

category: simplicial object