nLab
mate

Contents

Definition

Proposition

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.

a x a f λ f b y bb u a x a 1 a 1 ϵ f λ f η 1 b 1 b y b u a \array{ a & \overset{x}{\to} & a' \\ \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} \\ b & \underset{y}{\to} & b' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ b & \overset{u}{\to} & a & \overset{x}{\to} & a' & \overset{1}{\to} & a' \\ \mathllap{1} \downarrow & \mathllap{\epsilon} \Downarrow & \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} & \Downarrow \mathrlap{\eta'} & \downarrow \mathrlap{1} \\ b & \underset{1}{\to} & b & \underset{y}{\to} & b' & \underset{u'}{\to} & a' }

and

b y b u μ u a x aa f b y b 1 b 1 η u μ u ϵ 1 a 1 a x a f b \array{ b & \overset{y}{\to} & b' \\ \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} \\ a & \underset{x}{\to} & a' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ a & \overset{f}{\to} & b & \overset{y}{\to} & b' & \overset{1}{\to} & b' \\ \mathllap{1} \downarrow & \mathllap{\eta} \Uparrow & \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} & \Uparrow \mathrlap{\epsilon'} & \downarrow \mathrlap{1} \\ a & \underset{1}{\to} & a & \underset{x}{\to} & a' & \underset{f'}{\to} & b' }
Proof

That this is a bijection follows easily from the triangle identities.

Definition

The 2-cells λ\lambda and μ\mu in prop. 1 are called mates (or sometimes conjugates) with respect to the adjunctions fuf \dashv u and fuf' \dashv u' (and to the 1-cells xx and yy).

Properties

General

Proposition

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.

Proposition

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.

Naturality

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 cosntructed 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.

Examples

Example

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 familiar bijection

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

and the pasting operations as above yield the usual definition of the isomorphism of adjunction by means of unit and counit. Moreover, the naturality of the mate correspondence yields naturality of the bijection.

Example

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.

Example

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_! \,.

Ones 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.

References

Revised on March 9, 2014 20:09:10 by Mark Gomer? (68.54.30.66)