homotopy theory, (∞,1)-category theory, homotopy type theory
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A simplicial object $X$ in a category $C$ is an 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 space 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.
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.
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.
For $D$ a category, we write $D^{\Delta^{op}}$ for the functor category from $\Delta^{op}$ to $D$: its category of simplicial objects.
Let $D$ be a category with all limits and colimits. This implies that it is tensored over Set
This induces a functor
which we shall also write just “$\cdot$”.
For $X,Y \in D^{\Delta^{op}}$ write
and for $X,Y,Z \in D^{\Delta^{op}}$ let
be given in degree $n$ by
With the above definitions $D^{\Delta^{op}}$ becomes an sSet-enriched category which is both tensored as well as cotensored over $sSet$.
We may regard the category of cosimplicial objects $D^{\Delta}$ as an $sSet$-enriched category using the above enrichment by identifying
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:
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.
simplicial object
The original definition of simplicial objects, maps between them, and homotopies of such maps is due to Daniel M. Kan:
An early discussion of simplicial objects is in:
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
Last revised on May 23, 2022 at 14:35:00. See the history of this page for a list of all contributions to it.