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In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.

I know the definition (several of them), I've read the nlab page (and any good answers will be added there), I've worked with them, I've found examples of functors with and without adjoints, but I couldn't explain what an adjunction is to a five-year-old, the man on the Clapham omnibus, or even an advanced undergraduate.

So how should I intuitively think of adjunctions?

For more background: I'm a topologist by trade who's been learning category theory recently (and, for the most part, enjoying it) but haven't truly internalised it yet. I'm fully convinced of the value of adjunctions, but haven't the same intuition into them as I do for, say, the uniqueness of ordinary cohomology.

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Here's a related question on Maths-SX with a very nice answer in terms of posets: – Loop Space Sep 26 '11 at 8:52

The example I would give a five-year old is the following. Take the category R whose objects are real numbers (or perhaps rational numbers for the five-year old) and a single morphism x--> y whenever x is less than or equal to y. Let Z be the full subcategory consisting of the integers. The inclusion i:Z --> R has a left and a right adjoint: one is the floor function, the other the ceiling function. I think the five-year old will agree that these are approximations so I would then say that left and right adjoints are just jazzed up versions of approximations.

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OOPS - I forgot to post my name - Paul Smith – user1421 Jan 10 '11 at 15:10
That's going to be one amazing five-year-old. – KConrad Jan 10 '11 at 19:55
Actually, it's the other way around: the left adjoint is the ceiling and the right adjoint is the floor. If $r \in \mathbb{R}$ and $n \in \mathbb{Z}$ then $\lceil r \rceil \le n$ if and only if $r \le n$, and dually $n \le \lfloor r \rfloor$ if and only if $n \le r$. – Qiaochu Yuan Apr 7 '14 at 6:40
@QiaochuYuan could you expand why it's the other way around, if so, then suggest an edit to the answer? – CMCDragonkai Feb 17 at 7:53

I like Wikipedia's motivation for an adjoint functor as a formulaic solution to an optimization problem (though I'm biased, because I helped write it). In short, "adjoint" means most efficient and "functor" means formulaic solution.

Here's a digest version of the discussion to make this more precise:

An adjoint functor is a way of giving the most efficient solution to some optimization problem via a method which is formulaic ... For example, in ring theory, the most efficient way to turn a rng (like a ring with no identity) into a ring is to adjoin an element '1' to the rng, adjoin no unnecessary extra elements (we will need to have r+1 for each r in the ring, clearly), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.

The intuitive description of this construction as "most efficient" means "satisfies a universal property" (in this case an initial property), and that it is intuitively "formulaic" corresponds to it being functorial, making it an "adjoint" "functor".

In this asymmetrc interpretation, the theorem (if you define adjoints via universal morphisms) that adjoint functors occur in pairs has the following intuitive meaning:

"The notion that F is the most efficient solution to the (optimization) problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem which F solves."

Edit: I like the comment below emphasizing that an adjoint functor is a globally defined solution. If
$G:C\to D$, it may be true that terminal morphisms exist to some $C$'s but not all of them; when they always exist, this guarantees that they extend to define a unique functor $F:D\to C$ such that $F \dashv G$. This result could have the intuitive interpretation "globally defined solutions are always formulaic".

Compare this for example to the basic theorem in algebraic geometry that a global section (of the structure sheaf) of $\mathrm{Spec} (A)$ is always defined by a single element of $A$; the global sections functor is an adjoint functor representable by the formula $Hom(-,\mathrm{Spec}( \mathbb{Z}))$, so this is actually directly related.

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Just to emphasize on part of this response, note that an adjoint pair provides a global solution (category-wide) as opposed to a local solution provided by universal maps. So universality is more general, in that if the universal map exists for every object in the category, the family of universal maps can be used to define an adjoint. – marc Nov 23 '09 at 19:34
In the last section, should it be $\mathrm{Spec}(\mathbb{Z}[t])$ instead? – QcH Jan 2 '11 at 3:49

Suppose that $F\colon C\to D$ is a functor. Then there are many situations in which thinking of finding left and right adjoints to $F$ as solving approximation problems is very good intuition. So these would constitute functorial ways to approximate objects in $D$ relative to the image of $F$ by objects in $C$ either on the right or the left. I'm not sure I've really managed to word this in a way that conveys what I have in my head but here are some examples (which I have picked because they have a particularly 'approximationy' flavour but I do think that this works reasonably well in general anyway, I think my selection bias here is more skewed toward what I think about regularly).

Torsion theories are very good examples of this principal. For instance the notion of localization with respect to a homology theory in the homotopy category of spectra or more generally the approximating triangles coming from the acyclization and localization functors of a semi-orthogonal decomposition of a triangulated category. Another nice example along these lines is say the standard t-structure on the derived category of modules over some ring. Here we again have two pairs of adjoints and we can think of one as a right approximation by a bounded below complex coming from the unit and the other as a left approximation by a bounded above complex via the counit.

One can also view resolutions in the derived category in this way. For instance we have a right adjoint to the canonical map from $K(Inj R) \to D(R)$ for a ring $R$ where the first category is the homotopy category of complexes of injective $R$-modules which is taking K-injective resolutions. Similarly other sorts of resolutions, envelopes, and covers can be realised by adjunctions.

All of these examples are particularly nice in the sense that we get triangles or short exact sequences describing the object we start with in terms of our complementary approximations (by complementary I mean that there is orthogonality floating around in all of these examples so we have in some sense decomposed our category).

I think things like the adjoint functor theorem and Brown reprensentability become very reasonable from this point of view. One can loosely interpret them as saying provided things are "small enough" to be manageable and one has enough limits/colimits then one can build universal approximations (i.e. adjunctions) by taking coarse approximations and refining them.

I think this philosophy works well with the one given on the wiki page that Andrew Critch linked to.

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This intuition is the intuition behind Freyd's adjoint functor theorem. – Spice the Bird Jan 30 '13 at 16:07

Another intuitive notion for adjoint functors comes from the string diagram notation for $2$-categories. Functors are $1$-morphisms in the $2$-category $Cat$. Functors $D\xleftarrow L C$ and $C\xleftarrow R D$ are adjoint $L\dashv R$ iff there are natural transformations (i.e. 2-morphisms) $R\circ L\xleftarrow\eta 1_C$ and $1_D\xleftarrow\epsilon L\circ R$ such that the composites $$L\xleftarrow{\epsilon\circ 1_L}L\circ R\circ L\xleftarrow{1_L\circ\eta}L$$ $$R\xleftarrow{1_R\circ\epsilon}R\circ L\circ R\xleftarrow{\eta\circ 1_R}R$$ are equal to the identities $1_L,1_R$ respectively.

As mentioned before, this looks particularly nice in string diagram notation. A 1-morphism $B\xleftarrow f A$ is drawn as a vertical line labeled by $f$ bisecting a square with the left and right regions labeled by $B,A$ respectively. In the case that the 1-morphism is $1_A$ then you can remove the line and have a square labeled by $A$. Composition of $1$-morphisms $g\circ f$ is denoted by drawing 2 parallel vertical lines labelled $g,f$ respectively, trisecting a square, labeling regions according to source and targets of $C\xleftarrow g B\xleftarrow f A$ respectively. $2$-morphisms are drawn by putting a dot on a line bisecting a square. The dot is labeled by the $2$-morphism, the lower and upper lines it connects by its source and target $1$-morphisms respectively and the left and right regions by their source and target objects. Horizontal composition is drawn by trisecting the square as before and vertical composition by stacking your squares on top of each other. In the case that the $2$-cell is the identity $1_f$, it is drawn exactly like the $1$-cell $f$.

In general, a $2$-cell $g_1\circ\cdots\circ g_n\xleftarrow \varphi f_1\circ\cdots\circ f_m$ is drawn by having $n,m$ lines connecting the bottom or top of a square respectively to a dot labeled by $\varphi$ and labeling lines and regions accordingly. In the case of our $2$-morphism $R\circ L\xleftarrow\eta 1_C$, we have three lines emanating from the dot, however $1_C$ was to be drawn without a line so we have only a two lines connecting at a dot in the middle of the square. We drop the dot and simply draw it as a cup with appropriate labels. Similarly, we can draw $\epsilon$ as a cap. Finally, one can see the identity above is simply a planar isotopy of curves, something intuitive from topology.

See the catsters for a better explanation:

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For a "man on the Clapham omnibus" gloss on it, I think you could do worse than the Stanford Encyclopedia of Philosophy's entry for Category Theory. It describes adjoints as "conceptual inverses", and elaborates on how to see them that way in some of the standard examples.

I guess this is most probably a lower level than you were really asking for. But I think it articulates pretty well one of the less immediately obvious core points of the intuition (at least, my intuition) of what an adjunction is.

Putting this more precisely/abstractly: when we think of generalising isomorphism between objects of a 1-category to something between objects in a 2-category, we might usually think first of isomorphism and equivalence, but adjunction is also such a generalisation.

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