nLab mate





(mate bijection)
Given a 2-category KK, adjoint pairs (η,ϵ):fu:ba(\eta,\epsilon) \colon f \dashv u \colon b \to a and (η,ϵ):fu:ba(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a' , and 1-cells x:aax \colon a \to a' and y:bby \colon b \to b', there is a bijection

K(a,b)(fx,yf)K(b,a)(xu,uy) K(a,b')(f' x,y f) \cong K(b,a')(x u,u' y)

given by pasting with the unit of one adjunction and the counit of the other, i.e.:



That this is a bijection follows easily from the triangle identities, which say that the gray-shaded cells in the following pasting diagram cancel out;


The 2-cells λ\lambda and μ\mu in prop. are called mates [Kelly & Street (2006) p. 87; Leinster (2004), pp. 150] (earlier: conjugates, MacLane (1971), p. 98, see Exp. below) with respect to the adjunctions fuf \dashv u and fuf' \dashv u' (and to the 1-cells xx and yy).




Strict 2-functors preserve adjunctions and pasting diagrams, so that if F:KJF \colon K \to J is a 2-functor and if λ\lambda and μ\mu are mates wrt fuf \dashv u and fuf' \dashv u' in KK, then FλF \lambda and FμF \mu are mates wrt FfFuF f \dashv F u and FfFuF f' \dashv F u' in JJ.


If α:FG\alpha \colon F \Rightarrow G is a 2-natural transformation, then the naturality identities α bFf=Gfα a\alpha_b \circ F f = G f \circ \alpha_a and α aFu=Guα b\alpha_a \circ F u = G u \circ \alpha_b are mates wrt FfFuF f \dashv F u and GfGuG f \dashv G u.


There are two double categories with objects those of KK, vertical arrows adjoint pairs in KK and horizontal arrows 1-cells of KK. In one the 2-cells are those of the form λ\lambda above, while in the other they are those of the form μ\mu. It is easily shown, as in Kelly–Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any KK there is a double category Adj(K)Adj(K), defined up to isomorphism as above but with mate-pairs in KK as 2-cells.

What this means is that, for example, the mate of a square coming from a pasting diagram is given by pasting the mates of the individual 2-cells (whenever this makes sense).

In the double category Adj(K)Adj(K), every vertical arrow has both a companion (the left adjoint) and a conjoint (the right adjoint). (In fact, in some sense it is the universal double category constructed from KK with this property.) Therefore, it is equivalent to a 2-category equipped with proarrows. More explicitly, there is a forgetful functor L:Adj V(K)KL \colon Adj_V(K) \to K from the 2-category of objects, adjunctions and mate-pairs in KK to KK that sends an adjunction fuf \dashv u to ff. It is locally fully faithful, and moreover every LfL f has a right adjoint in KK by definition; this gives the more traditional definition of a proarrow equipment.



(conjugate transformation of adjoints)
Let FU:DCF \dashv U \colon D \to C be an adjunction in the 2-category Cat, i.e. a pair of adjoint functors, and A:*CA \colon * \to C and X:*DX \colon * \to D be objects of CC and DD considered as functors out of the terminal category **. Then taking mates with respect to 11:**1 \dashv 1 \colon * \to * and FUF \dashv U yields the hom-isomorphism

D(FA,X)C(A,UX) D(F A,\, X) \;\cong\; C(A,\, U X)

and the pasting operations as above yield the notion of conjugate transformation of adjoints. (This is the original notion, due to MacLane (1971), p. 98)

Moreover, the naturality of the mate correspondence yields naturality of the bijection.


If the ambient 2-category KK is the delooping of a monoidal category (𝒞,)(\mathcal{C}, \otimes) in that

KB 𝒞 K \simeq \mathbf{B}_\otimes \mathcal{C}

then an adjunction in KK is a pair of dual objects and the mate-construction is the construction of dual morphisms between dualizable objects.


Suppose given a commutative square (up to isomorphism) of functors:

f * g * k * h * \array{ & \overset{f^*}{\to} & \\ ^{g^*}\downarrow && \downarrow^{k^*}\\ & \underset{h^*}{\to} & }

in which f *f^* and h *h^* have left adjoints f !f_! and h !h_!, respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute

k *f *h *g * k^* f^* \to h^* g^*

has a mate

h !k *g *f ! h_! k^* \to g^* f_!

defined as the composite

h !k *ηh !k *f *f !h !h *g *f !ϵg *f !. h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_! \,.

One says that the original square satisfies the Beck-Chevalley condition if this mate is an equivalence.

Multi-variable mates

There is a version of the mate correspondence that applies to two-variable adjunctions and nn-variable adjunctions; see Cheng-Gurski-Riehl.

The relationship between two of the adjoints in a multivariable adjunction can be described as a parametrized adjunction: fixing the variables in each of the categories that appear in the domains of both adjoints, the pair of functors define an adjunction between the remaining two categories. Relative to the parametrized adjunctions that define a multivariable adjunction, the multivariable mates can be understood as parametrized mates.


The example of conjugate transformation of adjoints (but without the terminology of “mates”)

The explicit notion of mates may be officially due to

and is reviewed in:

Further review and discussion:

Discussion in the generalization of bicategories:

Last revised on December 4, 2023 at 11:21:19. See the history of this page for a list of all contributions to it.