Orthodox synthetic differential geometry (SDG) [1][2] is concerned with spaces of a special kind, called microlinear spaces. This paper is concerned with differential geometry of microlinear stacks or microlinear categories. It is exactly established in orthodox SDG that any infinitesimal transformation is invertible, that is to say, bijective. This paper shows that any infinitesimal endofunctor is invertible, as is expected. However, we are also expected to show that any infinitesimal natural transformation between infinitesimal endofunctors is invertible. Invertible natural transformations with respect to vertical composition of natural transformations are usually called natural isomorphisms in the literature. We will show that any infinitesimal natural transformation between infinitesimal endofunctors is invertible with respect to horizontal composition of natural transformations. We will show in § 3 that the category of infinitesimal endofunctors and infinitesimal natural transformations of a microlinear category is a categorical Lie algebra, just as the space of infinitesimal transformations of a microlinear space forms a Lie algebra in orthodox SDG.

The Ambrose-Palais-Singer theorem [3], claiming the equivalence of symmetric connections and sprays, is in essence of categorical nature. In orthodox SDG, it was first established by Bunge and Sawyer [4] under certain hypotheses, having been established in general by Kock and Lavendhomme [5]. We will establish the categorical Ambrose-Palais-Singer theorem in § 5, where the proof is none other than a paraphrase of Kock and Lavendhomme’s with strong emphasis on its categorical quintessence. § 4 addresses differential forms on the categorical level. We have confined our consideration to these three topics, but I expect that almost all the results of orthodox SDG can be elevated to the categorical level.

In universal algebra, an algebraic theory consists of a specification of operations and equational laws that these operations must satisfy. Models of the algebraic theory are called algebras. The notion of a monoid is a well-known example of an algebraic theory. A monoid is a set equipped with an associative binary operation and a unity element operating as both left and right unity. It is simple to categorify a given algebraic theory. By way of example, a categoricalmonoid is no other than an internal category within the category M o d of monoids, and their homomorphisms. Therefore, to make a category ℳ a categorical monoid, we must make both Ob   ℳ and Mor   ℳ monoids so that such categorical mappings as dom : Mor   ℳ → Ob   ℳ , cod : Mor   ℳ → Ob   ℳ , id : Ob   ℳ → Mor   ℳ and comp : Mor   ℳ × Ob   ℳ Mor   ℳ → Mor   ℳ are homomorphisms of monoids, i.e., mappings preserving binary operations and unity elements. Alternatively, a categoricalmonoid can be defined to be an internal monoid within the category C a t of small categories and functors. By way of example, a categorical monoid is a category ℳ equipped with functors F 2 : ℳ × ℳ → ℳ and F 0 : 1 → ℳ making the following diagrams commutative:

In general, any equational algebraic theory can be categorified by choosing functors corresponding to operations with commutative diagrams corresponding to equations.

The former approach is called the concept-first-then categorification approach, while the latter is called the category-first-then-conceptualization approach. It is well known that both approaches are essentially equivalent.

The setting in which we work is the same as that in [2]. In particular, R denotes the ring of line type which is supposed to abide by the general Kock-Lawvere axiom. We are slightly naive in referring to such a large category as C a t , but we are content only to say that there are several methods to make such discussions legitimate. We do not hope to blur our basic geometrical idea by such secondary legitimizations.

Every set can be regarded as a discrete category. We denote the category of categories and functors and that of sets and mappings by C a t and S e t , respectively. Sets and spaces are regarded as synonyms. Categories and stacks are regarded as synonyms. Vertical and horizontal compositions of natural transformations α , β are denoted by β ⋅ α and β ∘ α , respectively.

A natural transformation α : F ⇒ G is called a vertical natural isomorphism provided that there exists a natural transformation β : G ⇒ F such that α ⋅ β = id and β ⋅ α = id . This notion is simply called a natural isomorphism in the literature. A natural transformation α : F ⇒ G is called a horizontal natural isomorphism provided that there exists a natural transformation β : F ′ ⇒ G ′ such that α ∘ β = id and β ∘ α = id , which implies that all the compositions of functors F ′ ∘ F , F ∘ F ′ , G ′ ∘ G and G ∘ G ′ are the identity functors.

A functor F : ℳ → N is called isomorphic if there exists a functor G : N → ℳ such that G ∘ F = Id ℳ and F ∘ G = Id N .

It is well known that there is an adjunction

for ℒ , ℳ , N ∈ C a t , By taking ℒ to be a discrete category, i.e., a set S , we have

with ℳ × S = ∐ S ℳ and N S = ∏ S N . This means that C a t is tensored and cotensored over S e t .

An internal category ℳ within a complete category C is a pair ( Ob   ℳ , Mor   ℳ ) of objects in C along with morphisms in C

abiding by the usual axioms such as associativity of composition.

A ring of line type R is fixed. We recall some infinitesimal spaces.

The category of infinitesimal spaces, which is the dual of the category of Weil algebras, is of coproduct ⊕ and an initial object D 0 By way of example, we have

The following quasi-colimit diagrams will be used implicitly in frequence.

A category ℳ is called microlinear if ℳ D is a limit diagram in C a t for any finite quasi colimit diagram D of infinitesimal spaces. It is easy to see that

Proposition 2.1.Any limit of microlinear categories in C a t is microlinear. If ℳ is a microlinear category and N is a category, then ℳ N is a microlinearcategory. In particular,if ℳ is a microlinear category and S is a set, then ℳ S is a microlinear category.

Proof. The first statement follows from the interchangeability of two limits. The second statemnt follows from the exponential law ( ℳ N ) S = ( ℳ S ) N for any set S and the fact that the functor ( _ ) N : C a t → C a t preserves limits. □

Since the forgetful functor C a t → S e t × S e t preserves and creates limits, we have

Theorem 2.2.A microlinear category is no other than an internal category withinthe category of microlinear spaces and mappings.

In this paper, ℳ will always be a microlinear category.

We will show that the functor π 0 : ℳ D → ℳ assigning the object at 0 to each object in ℳ D and the morphism at 0 to each morphism in ℳ D a categorical R -modulefibered over ℳ in the sense that it is an internal R -module within the slice category C a t / ℳ .

Now we will define functors for a categorical R -module fibered over ℳ corresponding to scalar multiplication, addition, zero with respect to addition and inverse with respect to addition.

For each r ∈ R , the functor

is denoted by D r .

The functor

is denoted by D + , where the isomorphism ℳ D × ℳ ℳ D ≅ ℳ D ( 2 ) comes from the quasi-colimit diagram

The functor

is denoted by D 0 .

The functor

is denoted by D − .

Now we will show that the above four functors make ℳ D a categorical R -module fibered over ℳ .

Theorem 3.1.The functor π 0 : ℳ D → ℳ is a categorical R -module fibered over ℳ .

Proof. We have to make sure that the four functors corresponding to scalar multiplication, addition, zero with respect to addition and inverse with respect to addition abide by the usual equational axioms of R -module. Here we only deal with the associativity of addition, which follows from the commutativity of the following diagram:

□

Corollary 3.2.For any subcategory ℳ ′ ⊂ ℳ , the restriction of the functor π 0 to ( π 0 ) − 1 ( ℳ ′ ) → ℳ ′ is a categorical R -module fibered over ℳ ′ . In particular, for any object a ∈ ℳ , the category ( ℳ D ) a = ( π 0 ) − 1 ( { a , id a } ) is a categorical R -module.

Now we will show that the categorical R -module π 0 : ℳ D → ℳ fibered over ℳ is Euclidean in the sense that there is a unique functor ℳ D 2 ⇢ ℳ D making the diagram

commutative. Generally, a categorical R -module V is called Euclidean if there exists a unique functor φ : V D → V with

where π d is the standard projection V D → V assigning the value at d to each entity in V D . It is easy to see that a categorical R -module V is Euclidean if both the induced R -module Ob   V and the induced R -module Mor   V are Euclidean in the classical sense. It is also easy to see that if V is a Euclidean categorical R -module and N is a category, then V N is a Euclidean categorical R -module.

Theorem 3.3.The categorical R -module π 0 : ℳ D → ℳ fibered over ℳ is Euclidean.

Proof. This follows from the quasi-colimit diagram

and the commutativity of the diagram

Now we would like to make the categorical R -module ( ( ℳ ℳ ) D ) Id ℳ a categorical Lie R -algebra. It is easy to see that

Proposition 3.4.We have

for any ( F , G ) ∈ Ob ( ( ℳ ℳ ) D ) Id ℳ × ( ( ℳ ℳ ) D ) Id ℳ and any d ∈ D as well as

for any ( α , β ) ∈ Mor ( ( ℳ ℳ ) D ) Id ℳ × ( ( ℳ ℳ ) D ) Id ℳ and any d ∈ D , where ∘ stands for the composition of functors in the former formula, while ∘ stands for the horizontal composition of natural transformations in the latter formula.

Corollary 3.5.For any F ∈ Ob ( ( ℳ ℳ ) D ) Id ℳ and any d ∈ D , F d is an isomorphic endofunctor, while, for any α ∈ Mor ( ( ℳ ℳ ) D ) Id ℳ and any d ∈ D , α d is a horizontal natural isomorphism.

We define a functor D lie : ( ( ℳ ℳ ) D ) Id ℳ × ( ( ℳ ℳ ) D ) Id ℳ → ( ℳ ℳ ) D 2 by the formulas

where ∘ stands for the composition of functors in the former formula, while ∘ stands for the horizontal composition of natural transformations in the latter formula.

Proposition 3.6.There is a unique functor D Lie : ( ( ℳ ℳ ) D ) Id ℳ × ( ( ℳ ℳ ) D ) Id ℳ → ( ℳ ℳ ) D making the diagram

commutative.

Proof. This follows from the quasi-colimit diagram

and the fact that the diagram

is commutative. □

Now we have

Theorem 3.7. ( ( ℳ ℳ ) D ) Id ℳ is a categorical Lie R -algebra.

The proof is deferred to the Appendix.

In this section ℰ will always be a Euclidean categorical R -module which is microlinear as a category.

We define three functors:

(1) For any integer i between 1 and n and any r ∈ R , we define a functor D r i : ℳ D n → ℳ D n to be

(2) For any permutation σ of the numbers 1 , 2 , ⋯ , n , we define a functor D σ : ℳ D n → ℳ D n to be

(3) For any e ∈ D and any integer i between 1 and n + 1 , we define a functor F e i : ℳ D n + 1 → ℳ D n by

A categorical differential n -form on ℳ with values in ℰ is a functor ω : ℳ D n → ℰ abiding by the following two conditions:

(1) For any integer i between 1 and n and any r ∈ R , we have

(2) For any permutation σ of the numbers 1 , 2 , ⋯ , n , we have

where ε σ denotes the sign of the permutation σ .

Theorem 4.1.Let φ : ℳ D n × D n → ℰ be a functor abiding by the following conditions:

(1) For each ( e 1 , ⋯ , e n ) ∈ D n , the functor φ ( − , e 1 , ⋯ , e n ) : ℳ D n → ℰ is a categorical differential n -form on M with values in E ;

(2) For each f ∈ Mor   ℳ D n , the mapping φ ( f , − ) : D n → Mor   ℰ is homogeneous componentwise in the sense that

for each r ∈ R and each integer i between 1 and n .

Then there is a unique categorical differential n -form ω : ℳ D n → ℰ with values in ℰ such that

Proof. The proof is based on the quasi-colimit diagram

Since Ob   ℰ ℳ D n is a microlinear space and the diagram

equalizes for the mapping φ ¯ : D n → Ob   ℰ ℳ D n associated to φ , there exists a unique ω ¯ : D → Ob   ℰ ℳ D n with

Since Ob   ℰ ℳ D n is an Euclidean R -module, there is a unique ω ∈ Ob   ℰ ℳ D n with ω ¯ ( d ) = d ω for any d ∈ D . It is easy to see that is the desired categorical differential n -form on ℳ . □

Corollary 4.2.Given a categorical differential n -form ω with values in a categoricalEuclidean R -module ℰ , the functor ω ′ ˜ ℳ D n + 1 × D n + 1 → ℰ defined by the formula

abides by the three conditions in the above theorem, so that there exists a unique categorical differential ( n + 1 ) -form ω ′ with values in ℰ with

The corollary allows of exterior differentiation.

A categorical connection on ℳ is a functor

abiding by the following conditions:

(1) The diagram

is commutative, where π 1 : ℳ D × ℳ ℳ D → ℳ D is the forgetful functor to the first ℳ D .

(2) The diagram

is commutative, where π 2 : ℳ D × ℳ ℳ D → ℳ D is the forgetful functor to the second ℳ D .

(3) For any r ∈ R , the diagram

is commutative.

(4) For any r ∈ R , the diagram

is commutative.

A categorical connection ∇ : ℳ D × ℳ ℳ D → ℳ D 2 on ℳ is called symmetric if the diagram

is commutative, where σ is the permutation of 1 and 2.

A categorical spray on ℳ is a functor Δ : ℳ D → ℳ D 2 abiding by the following conditions:

(1) The diagram

is commutative, where i : D → D 2 is the natural injection inducing the functor ℳ i : ℳ D 2 → ℳ D .

(2) For any r ∈ R , the diagram

is commutative.

Proposition 5.1.For a categorical connection ∇ on ℳ , there is a unique functor △ ∇ : ℳ D → ℳ D 2 making the diagram commutativity of the diagram

commutative. The functor Δ ∇ is a categorical spray on ℳ .

Proof. The crucial part of the proof is based on the following quasi-colimit diagram:

owing to which, the unique functor Δ ∇ : ℳ D → ℳ D 2 characterized by the commutativity of the above diagram exists, because the diagram

is commutative, as can be seen easily. The funcor Δ ∇ : ℳ D → ℳ D 2 thus obtained is indeed a categorical spray on ℳ . □

Conversely, a categorical spray on ℳ determines a symmetric categorical connection on ℳ .

Proposition 5.2.For a categorical spray Δ on ℳ , there is a unique functor ∇ Δ : ℳ D × ℳ ℳ D → ℳ D 2 making the diagramgivecharacterized by the commutativity of the diagram

commutative. The functor ∇ Δ is a symmetric categorical connection ∇ Δ on ℳ .

Proof. The crucial part of the proof is based on the quasi-colimit diagram

owing to which, there is a unique functor ∇ Δ : ℳ D × ℳ ℳ D → ℳ D 2 making the above diagram commutative, because the diagram

is commutative. It is easy to see that the functor ∇ Δ : ℳ D × ℳ ℳ D → ℳ D 2 thus obtained is indeed a symmetric canonical connection on ℳ .

Theorem 5.3.For any categorical spray Δ on ℳ , we have

Proof. We will show that, for any symmetric categorical connection ∇ on ℳ , there is a unique functor Δ ∇ Δ ¯ : ℳ D → ℳ D 2 making the diagram

commutative. We simply juxtapose the commutative diagram characterizing ∇ Δ and the commutative diagram characterizing to get the commutative diagram