Contents

category theory

# Contents

## Idea

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.

## 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.

## 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.