Question about $\Delta(n)_{U}$ notaion in illusie's cotangent compelexe et deformations

In illusie's book cotangent complexe et deformations, 38page, the notation $\Delta(n)_{U}$ appears, and I cannot find the direct explanation or hint about meaning of this notation in this book.

I think $\Delta(n)$ means simply the simplicial set [0,n] in this book, but $\Delta(n)_{U}$ in 38page is different, I think It is an object of topos Simpl(T) but cannot understand what it exactly means.

Another question is also the question about notation in the same book.

In 56page of this book, the notation $\mathbb{Z}^{(\Delta(n))}$ appears, and I can't understand this notation and I can't find appropriate explanation in the book. In this case, the meaning of $\Delta(n)$ is not seams to be the only simplicial set [0,n]. I think it should be understanded by a sort of functor, but I can't find what it exactly means too.

If you knows about these notations in the book, or another resource explained about this part in english, it must be a big help to me. Thanks

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