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There are lots of reasonable definitions which only differ for $\kappa = \aleph_0$. However, I think what Lurie means is this: A $\kappa$-small simplicial set is a simplicial set with $< \kappa$ non-degenerate simplices. This is not equivalent to either of your definitions. For instance, if $\mathcal{C}$ is the category freely generated by an ...
Let $X$ be a polyhedron, and consider the inclusion $i : \operatorname{sk}_2 X \to X$. It induces a morphism on fundamental groups $i_* : \pi_1 \operatorname{sk}_2 X \to \pi_1 X$, for some choice of base point $x_0$ in the $0$-skeleton. This morphism is: Surjective: let $\alpha : I \to X$ be a representative of some class in $\pi_1 X$, $\alpha(0) = ... 2 We have $$H_0=\frac{\ker\partial_0}{\operatorname{im}\partial_1}.$$ Here,$\ker\partial_0$is the free$R$-module generated by$1,2,3,4$and$\operatorname{im}\partial_1$is the submodule of$\ker\partial_0$generated by$2-1,3-1$and$4-1$. Therefore the elements of the quotient have the form$\$\alpha ...