As far as I know, a simplicial k-chain is defined as $\sum c_i\sigma_i$, where $c_i$ are coefficients and $\sigma_i$ are k-simplices of a simplicial complex. So if I have a simplicial square defined by the 1-simplices $[0,1], [1,2], [2,3], [3,0]$, is $\omega = 1[0,1]+2[2,3]$ a 1-chain?
The definition tells me that it's a chain, but the terminology makes me believe that I'm missing something. Real chains are usually connected, so is the terminology misleading?