Lots of categories are contractible

This one is clear, but as a warm up here is a proof: there is a natural transformation from the identity functor to the constant functor with value the terminal object (whose components are the unique maps to the terminal object); the geometric realization of this natural transformation is a homotopy between the identity map and a constant map, and so the category is contractible.

Take any object \(X\) in the category \(C\). Consider the endofunctor \(F : C \to C\) given by \(FY = X \times Y\). Then (1) projection onto the first factor gives a natural transformation from \(F\) to the constant functor with value \(X\), and (2) projection onto the second factor gives a natural transformation from \(F\) to the identity.

Added [2017-07-09 Sun]: The earliest reference I have found for this result is Quillen’s wonderful Higher Algebraic K-theory: I.

Let’s show this in a very down to earth manner.

The more common definition says that a category is filtered if

1. for every pair of objects \(X\) and \(Y\) there are morphisms to a common object \(X \to Z\), \(Y \to Z\), and
2. for every pair of parallel morphisms \(f, g : X \to Y\) there is some morphism \(h : Y \to Z\) with \(hg = hf\).

Right away this tells us a filtered category is simply connected². Indeed, a) shows it is connected and b) shows that any pair of parallel arrows in \(C\) become equal in the groupoid \(C[C^{-1}]\). To show contractibility we’ll make use of an alternate characterization: a category is filtered if there is a cocone under any finite diagram in \(C\).³ I claim we can represent any class in \(\pi_n(BC)\) by a morphism of simplicial sets \(S \to NC\), where \(S\) is some simplicial set whose geometric realization is \(S^n\). Given this claim the proof is not hard: take a cocone under the diagram in \(C\) given by the image of the \(1\)-skeleton of \(S\); such a cocone let’s you extend the map \(S \to NC\) to a map \(\text{(co)cone}(S) \to NC\), showing the map we started with is inessential.

To prove the claim, we will use Kan’s \(\text{Ex}^\infty\), one construction for fibrant replacement in simplicial sets. Recall the basic facts:

One can define barycentric subdivision of a simplicial set analogously to the very classical barycentric subdivision of a simplicial complex, this a functor \({\text{Sd}}: {\text{sSets}}\to {\text{sSets}}\) which admits a right adjoint called \({\text{Ex}}\). Explicitly, the \(n\)-simplices of \({\text{Ex}}(X)\) are the maps \({\text{Sd}}(\Delta^n) \to X\). The important thing for us is that, while \({\text{Sd}}(\Delta^n)\) has the same geometric realization as \(\Delta^n\), it has “backwards” edges, and more generally simplices with “the wrong orientation”, meaning that a simplex in \({\text{Ex}}(X)\) is a kind of \(n\)-dimensional “zigzag” of simplices of \(X\). There is a natural inclusion \(X \subset {\text{Ex}}(X)\) (that just comes from a map \({\text{Sd}}(\Delta^n) \to \Delta^n\) – to figure out which map, remember that vertices of \({\text{Sd}}(\Delta^n)\) are subsets of \(\{0,1,\ldots,n\}\), the map we’re talking about sends a vertex of \({\text{Sd}}(\Delta^n)\) to its maximum element), and thus a whole chain of inclusions \(X \subset {\text{Ex}}(X) \subset {\text{Ex}}^2(X) \subset {\text{Ex}}^3(X) \subset \ldots\). The union of this chain is \({\text{Ex}}^\infty(X)\) and is a Kan complex whose geometric realization has the same homotopy type as the geometric realization of \(X\). Furthermore, any homotopy class of maps between the geoemtric realizations of two simplicial sets \(Y\) and \(X\), can be obtained by realizing a map of simplicial sets \(Y \to {\text{Ex}}^\infty(X)\) (this is part of what’s meant by the phrase “fibrant replacement”).

This means that any homotopy class in \(\pi_n(X)\) can be represented by a map \(\partial \Delta^n \to {\text{Ex}}^\infty(X)\), and since \(\partial \Delta^n\) has only finitely many non-degenerate simplices, this map must factor through some \({\text{Ex}}^k(X)\), which, by adjunction, means that the homotopy class we started with can be represented by a map of simplicial sets \({\text{Sd}}^k(\partial \Delta^n) \to X\), as required.

As pointed out to me by Clark Barwick, the answer is obviously no! Just consider any group as a one-object category.