amazing right adjoint


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Compact objects



William Lawvere’s definition of an atomic infinitesimal space is as an object Δ\Delta in a topos 𝒯\mathcal{T} such that the inner hom functor () Δ:𝒯𝒯(-)^\Delta : \mathcal{T} \to \mathcal{T} has a right adjoint (is an atomic object).

Notice that by definition of inner hom, () Δ(-)^\Delta always has a left adjoint. A right adjoint can only exist for very particular objects. Therefore the term amazing right adjoint.

Right adjoints to representable exponentials

Assume 𝒯=Sh(C)\mathcal{T} = Sh(C) is a Grothendieck topos, that the Grothendieck topology on the site CC is subcanonical. Let ΔCSh(C)\Delta \in C \hookrightarrow Sh(C) be a representable object. Then, () Δ:Sh(C)Sh(C)(-)^\Delta: Sh(C)\to Sh(C) has a right adjoint, hence Δ\Delta is an atomic infinitesimal space, precisely if () Δ:CC(-)^\Delta: C\to C preserves colimits. This is a special case of the general adjoint functor theorem. For if () Δ(-)^\Delta preserves colimits, its right adjoint is

() Δ:(YSh(C))(USh C(U Δ,Y)). (-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y)) \,.

The Y ΔY_\Delta defined this way is indeed a sheaf, due to the assumption that () Δ(-)^\Delta preserves colimits. So this is indeed a right adjoint.

  • A topos 𝒳\mathcal{X} is a local topos (over Set) if its global section functor Γ=Hom(1 𝒳,)\Gamma = Hom(1_{\mathcal{X}}, -) admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting 1 𝒳1_{\mathcal{X}} as “atomic”.

  • The topic of amazing right adjoints appears to have been studied mostly for toposes, but a related definition has been given for any category (with products): if OO is an exponentiable object of a category 𝒞\mathcal{C}, then OO being a tiny object in the sense of Definition 0.1 means that the endofunctor () O(-)^O has a right adjoint. (This adjoint is sometimes denoted () 1/O(-)^{1/O}, c.f. p. 269)

In this situation, one then has, symbolically

()×O() O() 1/O (-)\times O \quad \dashv\quad (-)^O \quad \dashv\quad (-)^{1/O}


The ubiquity of right adjoints to exponential functors in the context of synthetic differential geometry was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment lambda calculus with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997).

Revised on June 18, 2017 05:30:10 by Peter Heinig (