A paragraph in Mallat's "A wavelet tour of signal processing" says:
Spline Dyadic Wavelets A box spline of degree $m$ is a translation of $m+1$ convolutions of $\mathbf{1}_{[0,1]}$ with itself. It is centered at $t = \frac{1}{2}$ if $m$ is even and at $t=0$ if $m$ is odd.
Where $\mathbf{1}_{[0,1]}$ is the indicator function.
For $m=1$ I get:
\begin{eqnarray*} \phi(t) &=& \int_{-\infty}^{+\infty} \mathbf{1}_{[0,1]}(u) \mathbf{1}_{[0,1]}(t-u)du\\ &=& \int_0^1 \mathbf{1}_{[0,1]}(-(u-t))du \\ &=& \int_0^1 \mathbf{1}_{[-1,0]}(u-t)du \\ &=& \int_0^1 \mathbf{1}_{[t-1,t]}(u)du \end{eqnarray*}
The support of $\phi(t)$ is $[0,2]$, as far as I see. So, it's centered at $t=1$, not $t=0$. What am I doing wrong?
You are doing nothing wrong. Since one considers the whole family of integer shifted B-splines of the given order, one chooses the "generating" B-spline with center of mass closest to 0, thus the choice of 0 or 1/2. This is what the "is a translation" part of the definition is for.