ETAC

ETAC

0.1

Introduction

E

⁢

T

⁢

A

⁢

C

is the acronym for Lawvere’s

‘Elementary Theory of

Abstract Categories

’

which provides an axiomatic construction of the theory of

categories

and

functors

that was extended to
the

axiomatic theory of

supercategories

. The following

section

lists the

E

⁢

T

⁢

A

⁢

C

(or ETAC) axioms.

0.2

Axioms of

E

⁢

T

⁢

A

⁢

C

The ETAC axioms viz. (

[

6

]

) are :

0. For any letters

x

,

y

,

u

,

A

,

B

, and

unary function

symbols

Δ

0

and

Δ

1

,
and

composition

law

Γ

, the following are defined as

formulas

:

Δ

0

⁢

(

x

)

=

A

,

Δ

1

⁢

(

x

)

=

B

,

Γ

⁢

(

x

,

y

;

u

)

, and

x

=

y

; These formulas are to be, respectively, interpreted as
“

A

is the domain of

x

”, “

B

is the codomain, or range, of

x

”, “

u

is the composition

x

followed by

y

”,
and “

x

equals

y

”.

1. If

Φ

and

Ψ

are formulas, then “

[

Φ

]

and

[

Ψ

]

” , “

[

Φ

]

or

[

Ψ

]

”, “

[

Φ

]

⇒

[

Ψ

]

”, and “

[

n

⁢

o

⁢

t

⁢

Φ

]

” are also formulas.

2. If

Φ

is a formula and

x

is a letter, then “

∀

x

⁢

[

Φ

]

”,
“

∃

x

⁢

[

Φ

]

” are also formulas.

3. A string of symbols is a formula in ETAC iff it follows from the above axioms 0 to 2.

A

sentence

is then defined as any formula in which every

occurrence

of each letter

x

is within the scope of a

logical

quantifier

, such as

∀

x

or

∃

x

. The

theorems

of ETAC are defined as all those sentences which can be derived through logical inference from the following ETAC axioms:

4.

Δ

i

⁢

(

Δ

j

⁢

(

x

)

)

=

Δ

j

⁢

(

x

)

for

i

,

j

=

0

,

1

.

5a.

Γ

⁢

(

x

,

y

;

u

)

and

Γ

⁢

(

x

,

y

;

u

′

)

⇒

u

=

u

′

.

5b.

∃

u

⁢

[

Γ

⁢

(

x

,

y

;

u

)

]

⇒

Δ

1

⁢

(

x

)

=

Δ

0

⁢

(

y

)

;

5c.

Γ

⁢

(

x

,

y

;

u

)

⇒

Δ

0

⁢

(

u

)

=

Δ

0

⁢

(

x

)

and

Δ

1

⁢

(

u

)

=

Δ

1

⁢

(

y

)

.

6.

Identity

axiom:

Γ

⁢

(

Δ

0

⁢

(

x

)

,

x

;

x

)

and

Γ

⁢

(

x

,

Δ

1

⁢

(

x

)

;

x

)

yield always the same result.

7.

Associativity

axiom:

Γ

⁢

(

x

,

y

;

u

)

and

Γ

⁢

(

y

,

z

;

w

)

and

Γ

⁢

(

x

,

w

;

f

)

and

Γ

⁢

(

u

,

z

;

g

)

⇒

f

=

g

.
With these axioms in mind, one can see that

commutative diagrams

can be now regarded as certain

abbreviated

formulas corresponding to systems of equations such as:

Δ

0

⁢

(

f

)

=

Δ

0

⁢

(

h

)

=

A

,

Δ

1

⁢

(

f

)

=

Δ

0

⁢

(

g

)

=

B

,

Δ

1

⁢

(

g

)

=

Δ

1

⁢

(

h

)

=

C

and

Γ

⁢

(

f

,

g

;

h

)

, instead of

g

∘

f

=

h

for the arrows f, g, and h, drawn respectively between the
‘objects’ A, B and C, thus forming a ‘triangular commutative diagram’ in the usual sense of

category theory

. Compared with the

ETAC formulas

such diagrams have the advantage of a geometric–intuitive image of their

equivalent

underlying

equations

. The common property of A of being an object is written in shorthand as the abbreviated formula Obj(A) standing for the following three equations:

8a.

A

=

Δ

0

⁢

(

A

)

=

Δ

1

⁢

(

A

)

,

8b.

∃

x

[

A

=

Δ

0

(

x

)

]

∃

y

[

A

=

Δ

1

(

y

)

]

,

and

8c.

∀

x

∀

u

[

Γ

(

x

,

A

;

u

)

⇒

x

=

u

]

and

∀

y

⁢

∀

v

⁢

[

Γ

⁢

(

A

,

y

;

v

)

]

⇒

y

=

v

.

0.3

Remarks on ETAC interpretation

Intuitively, with this terminology and axioms a

category

is meant to be any

structure

which is a direct

interpretation

of ETAC. A

functor

is then understood to be a

triple

consisting of two such categories and of a rule F (‘the functor’) which assigns to each arrow or

morphism

x

of the first category,
a unique morphism, written as ‘

F

⁢

(

x

)

’ of the second category, in such a way that the usual two conditions on both objects and arrows in the standard functor definition are fulfilled (see for example

[

ICBM

]

)– the functor is well behaved, it carries object identities to image object identities, and commutative diagrams to image commmutative diagrams of the corresponding image objects and

image morphisms

. At the next level, one then defines

natural transformations

or

functorial morphisms

between functors as metalevel abbreviated formulas and equations pertaining to commutative diagrams of the distinct images of two functors acting on both objects and morphisms. As the name indicates natural transformations are also well–behaved, in terms of the ETAC equations being always satisfied.

References

1

R. Brown R, P.J. Higgins, and R. Sivera:

“

Nonabelian Algebraic Topology

:

Filtered Spaces

, Crossed Complexes, Cubical

Homotopy Groupoids

.”

EMS Tracts in Mathematics

, Vol.

15

, an EMS publication: September (2011), 708 pages. ISBN 978-3-13719-083-8.

2

R. Brown, J. F. Glazebrook and I. C. Baianu: A

Categorical

and

Higher Dimensional Algebra

Framework for

Complex Systems

and Spacetime Structures,

Axiomathes

17

:409–493, (2007).

3

I.C. Baianu:

Organismic Supercategories

: II. On Multistable Systems.

Bulletin of

Mathematical Biophysics

,

32

(1970), 539-561.

4

R. Brown and C.B. Spencer:

Double groupoids

and crossed modules,

Cahiers Top. Géom.Diff.

17

(1976), 343–362.

5

W.F. Lawvere: 1963. Functorial

Semantics

of Algebraic Theories.

Proc. Natl. Acad. Sci. USA

,

50

: 869–872.

6

W. F. Lawvere: 1966. The

Category of Categories

as a

Foundation

for Mathematics. , In

Proc. Conf. Categorical Algebra

–La Jolla, 1965, Eilenberg, S et al., eds. Springer–Verlag: Berlin, Heidelberg and New York, pp. 1–20.

Title

ETAC

Canonical name

ETAC

Date of creation

2013-03-22 18:16:15

Last modified on

2013-03-22 18:16:15

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

52

Author

bci1 (20947)

Entry type

Topic

Classification

msc 70A05

Classification

msc 60A05

Classification

msc 18E05

Classification

msc 55N40

Classification

msc 18-00

Synonym

axiomatic elementary theory of categories and functors

Synonym

ETAS sub-theory

Synonym

special case of ETAS

Synonym

axiomatic construction of the theory of categories and functors

Related topic

ETAS

Related topic

AxiomaticTheoryOfSupercategories

Related topic

FunctorCategories

Related topic

2Category

Related topic

CategoryTheory

Related topic

FunctorCategory2

Related topic

WilliamFrancisLawvere

Related topic

NaturalTransformationsOfOrganismicStructures

Defines

axiom of elementary theory of abstract categories

Defines

axiomatic theory of categories and functors

Defines

axiomatic construction of the theory of categories and functors

Defines

ETAC sentence

Defines

ETAC theorem

Defines

ETAC formula

Defines

Lawvere’s elementary theory of abstract categories

Defines

t

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