nLab adjoint (infinity,1)-functor theorem




The analogs in (∞,1)-category theory of the adjoint functor theorems in ordinary category theory.


  • Let G:𝒟𝒞G:\mathcal{D} \to \mathcal{C} be a functor between (,1)(\infty, 1)-categories. We say that GG satisfies the solution set condition if the (,1)(\infty, 1)-category c/Gc/G admits a small weakly initial set for any cc in 𝒞\mathcal{C}.

  • GG satisfies the hh-initial object condition if c/Gc/G admits an h-initial object for any cc in 𝒞\mathcal{C}.

  • 𝒞\mathcal{C} is said to be 2-locally small if for every pair of objects, xx,yy, of 𝒞\mathcal{C}, the mapping space map 𝒞(x,y)map_{\mathcal{C}}(x,y) is locally small.

  • Let G:𝒟𝒞G:\mathcal{D} \to \mathcal{C} be a continuous functor. Suppose that 𝒟\mathcal{D} is locally small and complete and 𝒞\mathcal{C} is 2-locally small. Then GG admits a left adjoint if and only if it satisfies the solution set condition.

  • Let G:𝒟𝒞G:\mathcal{D} \to \mathcal{C} be a finitely continuous functor?. Suppose that 𝒟\mathcal{D} is finitely complete. Then GG admits a left adjoint if and only if it satisfies the hh-initial object condition.


See Section 3 of (NRS18).

The following result is a consequence.


Let F:CDF : C \to D be an (∞,1)-functor between locally presentable (∞,1)-categories then

  1. it has a right adjoint (∞,1)-functor precisely if it preserves small colimits;

  2. it has a left adjoint (∞,1)-functor precisely if it is an accessible (∞,1)-functor and preserves small limits.


This is HTT, cor.


For the existence of right adjoints, we can weaken the hypotheses to merely requiring DD to be a locally small (infinity,1)-category. (HTT, rem.


  • Hoang Kim Nguyen, George Raptis, Christoph Schrade, Adjoint functor theorems for ∞-categories, Journal of the London Mathematical Society 101 2 (2019) 659-681 (arXiv:1803.01664)

Section 5.5 of

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