# Are connected categories with pullbacks weakly contractible?

Quillen's Theorem A says that a functor between (small) categories $f:I\rightarrow J$ induces a weak equivalence of the nerves if for each $j\in J$ the comma category $f/j$ is weakly contractible. In practice, one frequently shows a category is weakly contractible by showing it has an initial or terminal object, or that it is (co)-filtering. In the examples that keep coming up, the only obvious property is that the comma categories are connected and admit pullbacks.

Is this enough to show that these categories are weakly contractible? If not, is there a good counterexample?

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The category associated to the abelian group $\mathbb{Z}$? –  name Jul 3 '12 at 16:10
Or indeed, any non-trivial group works doesn't it? –  name Jul 3 '12 at 16:14
That is, take the category with one object $*$, and $hom(*, *) = G$ for some group $G$. A commutative square is just an identity $ab = cd$. For any pair of morphisms $a, b$, any commutative square (for example $a a^{-1} = b b^{-1}$) is a pull-back square. Now the geometric realisation of the nerve $NG$ is one way of constructing the classifying space of $G$ which has fundamental group $G$ by definition. Alternatively, you can show (see Weibel's book for example or probably May's book) simplicial sets of the form $NG$ are Kan, and then calculate directly that $\pi_1(NG) = G$. –  name Jul 3 '12 at 16:30
Perfect, and thanks. I should have worked through that example. –  Jesse Wolfson Jul 3 '12 at 18:54
If you had, then I wouldn't have. Thanks to you :) –  name Jul 4 '12 at 6:31