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Let ${\mathrm{tr}}_2$ be the truncation functor that takes a simplicial set and restricts it to dimensions at most 2. Its right adjoint is the 2-coskeleton functor. NLab says that the nerve of a small category is a 2-coskeletal simplicial set.

In a category, composition is defined only if two morphisms abut. Precisely, the composite $g\circ f$ is defined if the target of $f$ is the source of $g$. This translates to saying the nerve of a category is a simplicial set such that every inner horn of a 2-simplex has a filler.

My question is: Does this extension condition characterize those 2-coskeletal simplicial sets that is a nerve of a category? If not, is there a necessary and sufficient condition?

EDIT: Thanks Tyler for your comment. Can we identify a "category" with a "simplicial set truncated in dimension 2 such that every inner horn of a 2-simplex has a unique filler"? How does this account for the associativity of composition?

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The filler has to be unique. –  Tyler Lawson Sep 24 '12 at 4:50
ALL inner horns must be unique. –  Spice the Bird Sep 24 '12 at 5:53
An n-simplex has an inner horn only if n is at least 2. For a simplicial set truncated at dimension 2, "all inner horns" and "all inner horns of a 2-simplex" is the same thing. –  Colin Tan Sep 24 '12 at 9:28
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