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Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone category, is there some (and if so, what is the) appropriate category for which, given a functor with a left/right adjoint, we can characterize said adjoint as its initial/terminal object? My goal in this is to characterize [initial objects, universals, limits, adjunctions] all as examples of each other.

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  • $\begingroup$ it would be interesting to see all those connections in one place somewhere if you managed to write it down $\endgroup$
    – nicolas
    Commented Jan 16, 2016 at 8:39

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Well, yes: the left adjoint of a functor $G: C \to D$ is the initial object in the category whose objects are pairs $(H: D \to C, \eta: 1_D \to G H)$ where $\eta$ is a natural transformation, and whose morphisms $(H, \eta) \to (H', \eta')$ are natural transformations $\theta: H \to H'$ such that

$$\begin{array}{ccc} & 1_D & \\\\ {}^{ \eta} \swarrow & & \searrow {}^{\eta'} \\\\ GH & \xrightarrow{G\theta} & GH'\end{array}$$

commutes. Similarly, a right adjoint can be expressed as a terminal object in a suitable category (exercise in applying the concept of duality).

See the discussion on comma categories in Categories for the Working Mathematician,

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  • $\begingroup$ In other (but essentally the some) words, considering the lax-comma 2-category $CAt // D$ the left adjoint of $G$ is the initial object of the Hom-category $CAt // D[1_D, G]$. $\endgroup$
    – Buschi Sergio
    Commented Nov 25, 2012 at 8:45
  • $\begingroup$ Feel free to erase the commutative diagram if you don't like it. $\endgroup$
    – Martin Brandenburg
    Commented Nov 25, 2012 at 10:17
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    $\begingroup$ or in tom leinster book basic category theory, that's his 3rd presentation of adjunction $\endgroup$
    – nicolas
    Commented Jan 16, 2016 at 8:40
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    $\begingroup$ @Todd Trimble: If my understanding is correct, this defines the right Kan extension of identity functor of $D$ along $G.$ But in order to be a left adjoint, this Kan extension must be absolute. $\endgroup$
    – Bumblebee
    Commented Jun 21, 2020 at 17:48

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