nLab differential monad

Differential endofunctors and differential monads appear in the treatment of differential calculus on noncommutative spaces/schemes represented by their abelian categories “of quasicoherent sheaves”; the formalism is due Valery Lunts and Alexander Rosenberg.

Filtrations associated to topologizing subcategories

Below, a “subscheme” of an abelian category is in the sense of coreflective topologizing subcategory. A “subscheme” is Zariski closed if it is also reflective.

Let BB be an abelian category BB satisfying the Gabriel’s property sup. For any coreflective topologizing subcategory (“subscheme”) 𝕋\mathbb{T} of BB one defines the notion of 𝕋\mathbb{T}-filtration on any object MM of BB as an increasing M ={M i|i>1}M_\cdot = \{M_i \,|\, i\gt -1\} on MM such that M 1=0M_{-1} = 0 and M i/M i1Ob𝕋M_i/M_{i-1} \in Ob \mathbb{T}. By coreflectiveness of 𝕋\mathbb{T} there is a canonical choice of a 𝕋\mathbb{T}-filtration given by M i=𝕋 (i+1)MM_i = \mathbb{T}^{(i+1)}M (see n-th neighborhood of a topologizing subcategory); this is also called 𝕋 i+1\mathbb{T}^{i+1}-torsion (part) of MM. By the property (sup), there is a well-defined subobject 𝕋 ()(M):=sup{𝕋 (i)M}\mathbb{T}^{(\infty)}(M) := sup\{\mathbb{T}^{(i)}M\} which is calle the 𝕋\mathbb{T}-part of MM. An object is a 𝕋\mathbb{T}-object if it equals its own 𝕋\mathbb{T}-part. Denote by 𝕋 \mathbb{T}^\infty (note no brackets in the notation) the full subcategory of BB whose objects are 𝕋\mathbb{T}-objects.

Proposition. (Lunts-Rosenberg MPI1996-53, Lemma 3.3.1) The subcategory 𝕋 \mathbb{T}^\infty is a coreflective topologizing subcategory of BB.

The case of the diagonal “subscheme”: differential filtration

Fix an abelian category AA; let B=EndAB = End A be the category of additive endofunctors AAA\to A (essential smallness or some sort of accessibility assumption may be required on AA). If AA has the Gabriel’s property sup then EndAEnd A also has the property sup).

Let 𝕋\mathbb{T} be the diagonal subscheme of B=EndAB = End A, i.e. the minimal “subscheme” of the EndAEnd A containing the identity functor Id AId_A. For example, if AA is the category RMod{}_R Mod for RR a simple ring, then all “subschemes” of it are (Zariski) closed.

Definition. The Δ\Delta-part of any object MM of B=EndAB = End A is called the differential part of MM, and Δ\Delta-objects in BB are called differential endofunctors. Differential endofunctors are (Lunts-Rosenberg MPI 1996-53, 4.2) closed under composition. A differential monad is a monad whose underlying endofunctor is differential.

In these definitions it is often convenient to redefine BB as some other full subcategory BEndAB \subset End A containing Id AId_A, and closed under composition and colimits in EndAEnd A.

The case of (noncommutative) rings

If AA is the category RMod{}_R Mod and B=End cAB = End_c A of endofunctors having right adjoint for a (possibly noncommutative associative) kk-algebra RR, then this modification leads to the (noncommutative generalization of the) notion of differential bimodules. In other words, one looks at Δ R\Delta_R-part of bimodules where Δ R\Delta_R is (basically, by the Eilenberg-Watts theorem) equivalent to the full subcategory of the category of R kR opR\otimes_k R^{op}-modules (i.e. RR-bimodules) generated by all MM, such that the kernel of the multiplication is an ideal in R kR opR\otimes_k R^{op} contained in Ann(m)Ann(m).


The general abstract nonsense is proposed in these 1996 MPI preprints:

  • V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf

This is applied (without mentioning differential monads) to the case of noncommutative rings in

  • V. A. Lunts, A. L. Rosenberg, Differential operators on noncommutative rings, Selecta Math. (N.S.) 3 (1997), no. 3, 335–359 (doi); sequel: Localization for quantum groups, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123–159 (doi).

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