MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
it would be interesting to see all those connections in one place somewhere if you managed to write it down – nicolas Jan 16 at 8:39
up vote 15 down vote accepted

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,

share|cite|improve this answer
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 $CAt // D[1_D, G]$. – Buschi Sergio Nov 25 '12 at 8:45
Feel free to erase the commutative diagram if you don't like it. – Martin Brandenburg Nov 25 '12 at 10:17
I do like it -- thanks, Martin. – Todd Trimble Nov 25 '12 at 12:46
or in tom leinster book basic category theory, that's his 3rd presentation of adjunction – nicolas Jan 16 at 8:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.