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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 Set is a simplicial set.
A simplicial object in a category of presheaves is a simplicial presheaf.
A simplicial object in Top is a simplicial topological space.
A simplicial object in Diff is a simplicial manifold.
A simplicial object in the category Grp of groups is a simplicial group. See also Dold-Kan correspondence.
A simplicial object in the category of topological groups is a simplicial topological group.
A simplicial object in Lie algebras is a simplicial Lie algebra.
A simplicial object in Ring 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
simplicial object