A. Foundations and conventions
A.2. Categories
Our primary view on categories is the one which uses a family of hom-sets (cf. [42, Definition 1.2.1], [49, Section 1.1.1]).
Definition A.2.1. — A set (resp. class) C is acategory when there exist sets (resp.
classes)O,H,I, andVsuch that the following assertions hold:
a) C= (O,H,I,V)(where the ordered quadruple possibly needs to be interpreted in the class sense).
b) His a function onO×O.
c) Iis a function onOsuch that, for allx∈O, we haveI(x)∈H(x,x).
d) Vis a function onO×O×Osuch that, for all(x,y,z)∈O×O×O,V(x,y,z) is a function
V(x,y,z): H(y,z)×H(x,y)−→ H(x,z). e) (Units) For allx,y∈Oand all f ∈ H(x,y), we have:
(V(x,x,y))(f,I(x)) = f, (V(x,y,y))(I(y),f) = f. f) (Associativity) For allx,y,z,w∈Oand all
(h,g,f)∈ H(z,w)×H(y,z)×H(x,y), we have:
(V(x,y,w)) (V(y,z,w))(h,g),f
= (V(x,z,w)) h,(V(x,y,z))(g,f). From time to time, when we would like to stress the fact that a given category is either a set or a class, we use the following terminology: By asmall categorywe always mean a setCwhich is a category; by alarge categorywe mean a class—which might or might not be a set—Cwhich is a category.
A.2.2. WhenCis a category, then the four componentsC0,C1,C2, andC3are well-defined. Occasionally, we use the alternative denominations
ob(C):=C0, idC:=C2, homC:=C1, ◦C:=C3.
ob(C)is called theset(resp.class)of objects of C. homCis called thefamily of hom-sets of C. idCand◦Care called theidentity functionandcomposition function of C, respectively.
Sometimes we write sloppilyx∈Cinstead ofx∈C0. Sometimes we write sloppily C(x,y)instead ofC1(x,y). When we feel that confusion is unlikely to arise, we might omit the index referring to C in the expressions homC, idC, and ◦C. We use the customary infix notation
g◦f :=g◦Cf := (◦C(x,y,z))(g,f).
We say thatx is an object of/in Cwhenx ∈C0. We say that f is a morphism from x to y in Cwhenxandyare objects ofCand f ∈C1(x,y). Sometimes we paraphrase the latter statement by writing that “f: x→y is a morphism in C”.
A.2.3. The language of diagrams. — Very frequently throughout the text we will paraphrase certain (set-theoretic, categorical, or algebraic) statements by saying that a
“diagram commutes” in a given category. We exemplify this by looking at the picture
(or “diagram”):
(A.2.3.1) x f //
g
y
g0
x0
f0
//y0
Note that we do not formalize what a diagram is. Rather, we content ourselves with saying that a diagram is a picture (like (A.2.3.1), resembling a directed graph), in which one should be able to recognize a finite number ofverticesas well as a finite number of directed edges (orarrows) which are drawn between the vertices, i.e., for each arrow in the picture, it should be clear which of the vertices is the starting point and which of the vertices is the end point of the arrow. At each of the vertices in the picture we draw a symbolic expression corresponding to a term in our usual language.
The arrows may or may not be labeled by similar symbolic expressions.
LetCbe a category (class or set). Then we say thatthe diagram(A.2.3.1)commutes in Cwhen the following assertions hold:
a) x,y,x0,y0are objects ofC.
b) We have f ∈C1(x,y),g∈C1(x,x0),g0∈C1(y,y0), and f0∈C1(x0,y0). c) We have
(◦C(x,y,y0))(g0,f) = (◦C(x,x0,y0))(f0,g).
From this we hope it is clear how to translate the phrase “the diagram . . . commutes inC” into a valid formula of our set theoretic language for an arbitrary diagrammatic picture in place of the dots. Notice that the above assertions a) and b) correspond respectively to the facts that (the labels at) the vertices of the diagram are objects ofCand that, for every ordered pair of vertices of the diagram, every (label at an) arrow drawn between the given vertices is a matching morphism inC. So to speak, assertions a) and b) taken together say that the picture (A.2.3.1)is a diagram in C. The final assertion c) says that the diagram is actuallycommutative in C.
Observe that, strictly speaking, the phrase “the diagram . . . commutes inC” makes sense only if all the arrows in the given diagram are actually labeled. Therefore, when we say the phrase referring to a diagram with some arrows unlabeled, we ask our readers to kindly guess the missing arrow labels from the individual context. Usually, in this regard, an unlabeled arrow corresponds to some sort of canonical morphism.
At times, we will stress this point by labeling the arrow with a “can”.
Sometimes the concrete arrow labels (or corresponding morphisms) rendering a given diagram commutative in a category are irrelevant. In these cases we use the phrase “there exists a commutative diagram . . . in C”. For instance, when we say there
exists a commutative diagram
x //
f
&&
y //z
inC, we mean that there exist f0and f00such that the diagram
x f0
//
f
&&
y
f00
//z
commutes inC(in particular, we see thatx,y, andzneed to be objects ofCand f0 needs to be an element ofC1(x,y), etc.).
In some cases, where we want to draw attention to specific arrows in a diagram—
mostly this happens when we assert that the morphisms corresponding to the arrows exist or exist in a unique way—, we print these arrows dotted as in:
x f //y
Statementwise, the dotted style of an arrow has no impact whatsoever.
Occasionally, instead of an ordinary arrow we will draw a stylized, elongated equality sign as, e.g., in:
x y
These “equality sign arrows” go without label. Then, to the interpretation of the discussed commutativity statements, we have to add the requirement thatx = y;
moreover, in order to check the actual commutativity of the diagram (cf. assertion c) above), one simply merges (successively) any two vertices in the diagram which are connected by an equality sign arrow into a single vertex until one ends up with an equality sign free diagram. As an alternative, one replaces each occurrence of an equality sign arrow with an ordinary arrow, choosing the direction at will, and attaches the label idC(∗)to it, where∗is to be replaced by the label at the chosen starting point. So, the previous diagram becomes either
x idC(x) //y or x oo idC(y) y
Last but not least, we occasionally draw “∼” signs at arrows in diagrams (possibly in addition to already existing labels), like, for instance, in:
x f
∼ //y
In these cases, we add the requirement that f be an isomorphism fromxtoyinCto the interpretation of any commutativity statement.
A.2.4. We use the following notation for the “standard categories”, where the respec-tive rigorous definitions are to be modeled after Definition A.2.1: