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I am using the definition of the simplex category from N-Lab:

Definition 1($\Delta$): The path categories on finite directed acyclic graphs with at least one vertex are a collection of objects in CAT. The full subcategory generated by these objects and the functors between them is $\Delta$.

The $n-lab$ article refers to picking a representative in each isomorphism class of objects in $\Delta$ and I don't understand what this means, hence my question.

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    The definition you gave is none of the definitions on that page. – 2017-01-21
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    I tried to give the definition of the simplex category in one go in terms that I understood. – 2017-01-21
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    The definition is the same except for the fact that I substituted linear directed graph, with directed acyclic graph, and free category with path category. (But please correct me if in doing this I gave an inequivalent definition! That is why I am here). – 2017-01-21
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    Okay. I might recommend rephrasing the question to "is this definition equivalent?". Or, if you aren't really concerned by that, pointing out what parts of the definition on the nLab page you find confusing (which, admittedly, you've partially done). – 2017-01-21
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    To partially answer your question, the objects of $\Delta$ are categories, so the arrows are functors not natural transformations. Further, a collection of vertices with no edges is a directed acyclic graph but not a linear one (if there's more than one vertex) and does lead to a very different category. – 2017-01-21
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    Thank you. Is a directed linear graph the same thing as a connected directed acyclic graph?(I am asking because google only gave directed acyclic graphs when I searched.) – 2017-01-21
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    No, connected directed acyclic graphs are much more general. You can find many examples at https://en.wikipedia.org/wiki/Directed_acyclic_graph. – 2017-01-21
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    A linear directed graph is essentially just a list of vertices (without duplicates) with an edge from each element in the list to the next which we can leave implicit. It's a bit silly, but you could call it a "unary tree", it certainly is a tree in the graph theory sense. At any rate, it's easier to understand "linear directed graph" directly than as a directed acyclic graph with some special properties. Indeed, the equivalent description from that page as "finite totally order sets" is simple and captures it well. – 2017-01-21
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    The only reason I went for this alternative viewpoint is because Paul Goerss's notes described the degeneracy and inclusion of faces maps, by functors from $[n-1] \to [n]$ and $[n+1] \to [n]$. To make sense of this, one has to view $[n]$ as a category and the most straightforward way of doing this seemed to be to view $[n]$ as the path category on the graph $0 \to 1 \to 2 \to ...n$. – 2017-01-21
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    Yes, that is how you're supposed to view it, as the path category on that graph, which is linear. Or, as the totally ordered set indicated by the same picture, with its usual category structure. – 2017-01-22

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