We study new categories of ’ $C^\infty$-schemes with corners’, following the definition of schemes in Algebraic Geometry. A ‘(local) $C^\infty$-ringed space with corners’ is a topological space with a sheaf of $C^\infty$-rings with corners. We define a spectrum functor from $C^\infty$-rings with corners to local $C^\infty$-ringed spaces with corners, adjoint to the global sections functor. Then a $C^\infty$-scheme with corners is a local $C^\infty$-ringed space with corners which is locally isomorphic to the spectrum of a $C^\infty$-ring with corners.

Manifolds with (g-)corners embed into $C^\infty$-schemes with corners as full subcategories. Thus, $C^\infty$-schemes with corners are a vast generalization of manifolds with corners and manifolds with g-corners.

In ordinary Algebraic Geometry, the spectrum functor is full and faithful, but for $C^\infty$-rings with corners neither holds. ‘Semi-complete’ $C^\infty$-rings with corners are a special class on which the spectrum functor is faithful and surjective, but not full.

We study special classes of $C^\infty$-schemes with corners, and existence of fibre products of $C^\infty$-schemes with corners.

We discuss four generalizations and applications of $C^\infty$-schemes with corners.

1. 1. To provide a model for a theory of ‘Synthetic Differential Geometry with corners’.

2. 2. To a theory of ‘ $C^\infty$-stacks with corners’. $C^\infty$-Stacks are studied in D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. Most of the theory extends to corners with no changes.

3. 3. To a theory of ‘ $C^\infty$-schemes with a-corners’. ‘Manifolds with a-corners’ are introduced in D. Joyce, arXiv:1605.05913 as a class of manifolds with corners with an alternative smooth structure, that has applications in analysis, e.g. Morse theory moduli spaces should really be manifolds with a-corners, not corners.

4. 4. To a theory of ‘derived $C^\infty$-schemes and derived $C^\infty$-stacks with corners’, and within these, ‘derived manifolds with corners’ and ‘derived orbifolds with corners’, where ‘derived’ is in the sense of Derived Algebraic Geometry.

If X is a manifold then the set $C^\infty$(X) of smooth functions on X is a ’ $C^\infty$-ring’, a rich algebraic structure with many operations. ‘ $C^\infty$-schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to ‘derived manifolds’ in Derived Differential Geometry.

Manifolds with corners, such as a triangle or a cube, are generalizations of manifolds, with boundaries and corners. They occur in many places in Differential Geometry. In this book we define and study new categories of ‘ $C^\infty$-rings with corners’ and ‘ $C^\infty$-schemes with corners’, which generalize manifolds with corners in the same way that $C^\infty$-rings and $C^\infty$-schemes generalize manifolds. These will be used in future work by the second author as the foundations of theories of derived manifolds and derived orbifolds with corners. These have important applications in Symplectic Geometry, as moduli spaces of pseudo-holomorphic curves should be derived orbifolds with corners.

We define and study new categories of ‘pre $C^\infty$-rings with corners’ and ‘ $C^\infty$-rings with corners’. These are pairs (C,C_{ex}) of sets with many algebraic operations. If X is a manifold with corners then taking C to be the set of smooth maps X -> R, and C_{ex} the set of smooth maps X -> [0,infinity), gives a $C^\infty$-ring with corners. It helps to consider C a $C^\infty$-ring or R-algebra, and C_{ex} a commutative monoid which adds ‘corner structure’ to C.

A (pre) $C^\infty$-ring with corners can equivalently defined as a product-preserving functor from the category of Euclidean spaces with corners [0,infinity)^k x R^{n-k} to the category of sets. This is an example of an Algebraic Theory, so theorems from Algebraic Theories apply.

We define interesting subcategories of $C^\infty$-rings with corners with special properties - interior, firm, toric, ... and study their categorical properties, e.g. existence of (co)limits.

Much of the theory of commutative algebras and $C^\infty$-rings used in scheme theory extends to $C^\infty$-rings with corners, e.g. local $C^\infty$-rings with corners and localization, free $C^\infty$-rings with corners, generators, and relations.

A ‘manifold with corners’ is a space locally modelled on [0,infinity)^k x R^{n-k}, just as manifolds are locally modelled on R^n. Triangles, squares and cubes are examples. A manifold with corners X has a boundary dX, a manifold with corners of dimension one less. The boundary of a triangle is the three edges. If f : X -> Y is a smooth map of manifolds with corners then f need not map dX -> dY, i.e. boundaries are not ‘functorial’. But we can extend the boundary to the ‘corners’ C(X), a manifold with corners of mixed dimension, which is functorial, i.e. f extends to C(f) : X -> Y.

There are several notions of smooth map of manifolds with corners. We choose the ‘b-maps’ of Richard Melrose. There are two notions of tangent bundle, the ordinary tangent bundle TX, functorial under smooth maps, and the ‘b-tangent bundle’ bTX.

‘Manifolds with g-corners’ are a generalization with exotic corner structure. They have some nicer properties, e.g. existence of b-transverse fibre products.

Manifolds with corners occur in many places. e.g. in TQFTs, and analysis of partial differential equations. Some moduli spaces in Morse theory, Floer theories, and Symplectic Geometry are manifolds with corners.

A module over a $C^\infty$-ring, or $C^\infty$-ring with corners, is a just a module over the underlying R-algebra. Similarly, a sheaf of modules over a $C^\infty$-scheme, or $C^\infty$-scheme with corners, is a sheaf of modules over the underlying sheaf of R-algebras. So the general theory of modules, sheaves of modules, spectrum functors for modules, and (quasi-)coherent sheaves, extends from ordinary Algebraic Geometry with little extra work.

A manifold with corners has two notions of (co)tangent bundle: the (co)tangent bundles TX, T*X, independent of the corners, and the b-(co)tangent bundles bTX, bT*X, depending on the corners. Similarly, a $C^\infty$-ring with corners has two cotangent modules: the cotangent module, which depends only on the underlying $C^\infty$-ring, and the b-cotangent module, which also depends on the corner structure. Thus a $C^\infty$-scheme with corners has a cotangent sheaf and b-cotangent sheaf.

If X is a manifold with corners, regarded as a $C^\infty$-scheme with corners, the (b-)cotangent sheaf is the sheaf of sections of the (b-)cotangent bundle.

(b-)Cotangent sheaves are covariantly functorial, and have exact sequences for fibre products.

This chapter summarizes D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. If X is a manifold then the set $C^\infty$(X) of smooth functions on X is a ‘ $C^\infty$-ring’, a rich algebraic structure with many operations. A module over a $C^\infty$-ring is a module over it as an real algebra. A $C^\infty$-ring has a cotangent module. For $C^\infty$(X) this is the sections of the cotangent bundle T*X.

‘ $C^\infty$-Schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry, and of allowing Differential Geometers to study spaces far more general than manifolds. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to ‘derived manifolds’ in Derived Differential Geometry. The category of $C^\infty$-schemes has good properties: it is Cartesian closed and has all finite limits and directed colimits.

We also study sheaves of modules over $C^\infty$-schemes, as for (quasi-)coherent sheaves in Algebraic Geometry. A $C^\infty$-scheme has a cotangent sheaf, generalizing the cotangent bundle of a manifold.

If X is a manifold with corners of dimension n, the boundary dX is a manifold with corners of dimension n-1, and the k-fold boundary d^kX a manifold with corners of dimension n-k. The ‘k-corners’ C_k(X) is d^kX/S_k, also a manifold with corners of dimension n-k. The ‘corners’ C(X) is the disjoint union of all C_k(X), a manifold with corners of mixed dimension.

A smooth map f : X -> Y of manifolds with corners need not map dX -> dY, that is, boundaries are not functorial. But there is a natural map C(f) : C(X) -> C(Y), which need not map C_k(X) -> C_k(Y). This is the ‘corner functor’ for manifolds with corners.

We extend the corner functor to $C^\infty$-schemes with corners. It is right adjoint to the inclusion functor from interior $C^\infty$-schemes with corners to all $C^\infty$-schemes with corners, and so is canonically determined by the notion ‘interior’. Using the corner functor we define boundaries and corners of ‘firm’ $C^\infty$-schemes with corners.

We use the corner functor to study fibre products of $C^\infty$-schemes with corners, and show that b-transverse fibre products of manifolds with (g-)corners map to fibre products of $C^\infty$-schemes with corners.