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    • CommentRowNumber1.
    • CommentAuthorAli Caglayan
    • CommentTimeMar 18th 2019
    • (edited Mar 18th 2019)

    In category theory, a functor U:CDU: C \to D can have the property of having a left adjoint iff XD\forall X \in D, FXC\exists FX \in C and η X:XU(FX)\exists \eta_X : X \to U(F X) such that, AC\forall A \in C and forall f:XU(A)f : X \to U(A), !g:FXA\exists ! g : F X \to A such that the triangle commutes: f=U(g)η Xf = U(g) \circ \eta_X.

    This definition is nice because it is very minimalistic, and it can be canonically extended to construct a left adjoint functor when it holds. I suppose another way to say this is that these are the necessary and sufficient conditions for a left adjoint defined any other way.

    My question is: Do we have a similar minimal sort of definition for having an oo-left adjoint?

    I don’t have any particular model of oo-cat in mind.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 18th 2019

    The answer is yes. But in order to make it precise one probably has to choose a particular model.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 18th 2019

    The first functor should probably be called U, not F. Also A∈C, not A∈X.

    The condition after “such that” can be reformulated by saying that the canonical functor FX/C→X/U induced by U and precomposition with η_X is an equivalence of categories.

    Presumably, replacing equivalences with ∞-equivalences should yield the desired outcome.

    • CommentRowNumber4.
    • CommentAuthorAli Caglayan
    • CommentTimeMay 26th 2019

    So I had a chat with Emily and it turns out there is a way to make this precise:

    A functor U:CDU : C \to D has a left adjoint if for all XX in DD, the comma category X/UX / U has an initial object (FX,η X)(FX, \eta_X).

    Then you should be able to stick an oo infront of everything.

    This is proposition 4.1.5 in her new book.

    • CommentRowNumber5.
    • CommentAuthorAli Caglayan
    • CommentTimeMay 26th 2019

    Am I correct in thinking that because we are considering oo-initial objects, we typically don’t even need to consider higher coherences?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2019

    That probably depends on your definition of \infty-initial…