ETAC 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 ETAC axioms are listed next.
0. For any letters
, and unary function symbols
and
,
and composition law
, the following are defined as formulas:
,
,
, and
; These formulas are to be, respectively, interpreted as
“
is the domain of
", “
is the codomain, or range, of
", “
is the composition
followed by
",
and “
equals
".
1. If
and
are formulas, then “
and
” , “
or
”, “
”, and “
” are also formulas.
2. If
is a formula and
is a letter, then “
”,
“
” 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
is within the scope of a quantifier, such as
or
. The theorems of ETAC are defined as all those sentences which can be derived through logical inference from the following ETAC axioms:
4.
for
.
5a.
and
.
5b.
;
5c.
and
.
6. Identity
axiom:
and
yield always the same result.
7. Associativity axiom:
and
and
and
.
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:
,
,
and
, instead of
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.
,
8b.
,
and
8c.
and
.
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
of the first category,
a unique morphism, written as `
' 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 [])- 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 satisfied.
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