pure morphism (much more to be said, and more references, but no time now)

]]>I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the infinitesimal commutator, and that the functor expressed by this operation factors through formal group laws (FGLs) in the usual way. This reveals that Lie groups are FGLs with respect to first-order infinitesimals.

Now I would like to consider a lined topos equipped with higher-order infinitesimals, and develop in this context a modified notion of microlinearity. I have not yet developed the details of this. But does modifying microlinearity in this way, to yield R-modules by exponentiating FGLs with higher-order infinitesimals, sound reasonable? It is worth saying that in general we want certain polynomial identities to hold in the resulting R-modules, e.g. the Jacobian identity.

While FGLs have been thought of in this way (e.g. Didry in [1], an attempt to extend Lie theory to include Leibniz algebras), I have not found sources discussing modifications of microlinearity to subsume FGLs in the language of SDG. Some suggestive remarks can be found in Nishimura’s work, such as in the introduction of the paper [2], where the author discusses prolongations of spaces with respect to polynomials algebras as generalizations of Weil algebras. What do you think, nForum?

[1] Didry, M. Construction of Groups Associated to Lie- and to Leibniz- Algebras

[2] Nishimura, H. Axiomatic Differential Geometry II-2, Chapter 2: Differential Forms

]]>**Changes-note**. Changed the already existing page 201707071634 to now contain a different svg illustration, planned to be used in an integrated way in pasting schemes soon.

**Metadata.** Like here, except that in 201707071634 symbols (arrows) indicating what is to be interpreted to 2-cells are given, in the same direction as in Power’s paper.

At scheme, the definition of a $k$-ring notates the category as $k/Ring$ and says it’s a pair $(R,f\colon k\to R),$ with $f$ a $k$-algebra homomorphism. Is this correct? What does this mean? We can obviously view $R$ as a $k$-algebra *by means of the action by $f$*. How can we say that $f$ is itself a $k$-linear map? With respect to what $k$-action on $R$? It’s somehow circular.

Perhaps does it mean to say something more like “a $k$-ring is an object in the undercategory $k/Ring$, so objects are *all* pairs $(R,f\colon k\to R)$ (no restriction on $f$), and morphisms are $k$-algebra homomorphisms”?

New entry affine morphism (redirecting also affine morphism of schemes) and a related post at MathOverflow, linked there. Expansion of the material at affine scheme including few words on relative affine schemes and on fundamental theorem on morphisms of schemes.

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