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As written, the answer would trivially be that this is impossible.
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Tim Campion
Tim Campion
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Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not Joyal weakly equivalent to a 2-coskeletal simplicial setquasicategory? (Equivalently, mapping simplicial sets between objects of $Y$ have nontrivial homotopy groups in degree 1 or higher.)

If $X$ satisfies the Segal conditions, then $X$ is the nerve of a 1-category, hence is Joyal fibrant, so such $X$ cannot be an example.

In the Kan model structure on simplicial sets, examples are easy to construct: the Kan fibrant replacement of the nerve of the delooping of a monoid $M$ is the homotopy group completion of $M$, which can have nontrivial higher homotopy groups.

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not Joyal weakly equivalent to a 2-coskeletal simplicial set? (Equivalently, mapping simplicial sets between objects of $Y$ have nontrivial homotopy groups in degree 1 or higher.)

If $X$ satisfies the Segal conditions, then $X$ is the nerve of a 1-category, hence is Joyal fibrant, so such $X$ cannot be an example.

In the Kan model structure on simplicial sets, examples are easy to construct: the Kan fibrant replacement of the nerve of the delooping of a monoid $M$ is the homotopy group completion of $M$, which can have nontrivial higher homotopy groups.

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not Joyal weakly equivalent to a 2-coskeletal quasicategory? (Equivalently, mapping simplicial sets between objects of $Y$ have nontrivial homotopy groups in degree 1 or higher.)

If $X$ satisfies the Segal conditions, then $X$ is the nerve of a 1-category, hence is Joyal fibrant, so such $X$ cannot be an example.

In the Kan model structure on simplicial sets, examples are easy to construct: the Kan fibrant replacement of the nerve of the delooping of a monoid $M$ is the homotopy group completion of $M$, which can have nontrivial higher homotopy groups.

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Dmitri Pavlov
Dmitri Pavlov
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Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not Joyal weakly equivalent to a 2-coskeletal simplicial set? (Equivalently, mapping simplicial sets between objects of $Y$ have nontrivial homotopy groups in degree 1 or higher.)

If $X$ satisfies the Segal conditions, then $X$ is the nerve of a 1-category, hence is Joyal fibrant, so such $X$ cannot be an example.

In the Kan model structure on simplicial sets, examples are easy to construct: the Kan fibrant replacement of the nerve of the delooping of a monoid $M$ is the homotopy group completion of $M$, which can have nontrivial higher homotopy groups.