As discussed here, Using the universal property of spaces, the $(\infty,1)$-category of spaces has a universal property: it is the free $\infty$-categorical cocompletion of the terminal category $*$. That is, for any $(\infty,1)$-category $C$ with all colimits, there is an equivalence $$ Fun^{cocontinuous}(Spaces, C) \cong Fun(*,C) \cong C $$ I am wondering if there is a similar universal property for $Cat^{(\infty,1)}$, the $(\infty,2)$-category of $(\infty,1)$-categories.

Note that $Cat^{(\infty,1)}$ has all lax colimits and any category $C$ is the lax colimit of the functor $C \rightarrow * \rightarrow Cat^{(\infty,1)}$. So one guess is that $Cat^{(\infty,1)}$ is the lax cocompletion of the terminal category $*$ in the sense that for any $(\infty,2)$-category $C$ with lax colimits, there is an equivalence $$ Fun^{Lax \ cocontinuous}(Cat^{(\infty,1)}, C) \cong Fun(*, C) \cong C $$ Does this proposed equivalence hold and, if so, has it been studied somewhere?

  • 2
    $\begingroup$ "the category of spaces" here being the $(\infty,1)$-category of (homotopy types of) spaces, and cocompletion is homotopy cocompletion. Obviously you know this, but leading with the abuse of terminology is not helpful. $\endgroup$
    – David Roberts
    Commented Jul 10 at 3:17
  • 4
    $\begingroup$ There should be a result along the lines that if $V$ is a cocomplete closed symmetric monoidal ($\infty$?)-category then $V$ is the free cocomplete $V$-enriched category on a point, a special case of the universal property of presheaf categories. Morally we'd like to apply that result to $V = \text{Cat}$ here. $\endgroup$ Commented Jul 10 at 3:57


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.