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1 Morphisms and objects
Some morphisms and some objects are special enough to garner special names that we will use regularly.
In morphisms, the important properties are
- cancellability - the categorical notion corresponding to properties we use when solving, e.g., equations over :
- existence of inverses - which is stronger than cancellability. If there are inverses around, this implies cancellability, by applying the inverse to remove the common factor. Cancellability, however, does not imply that inverses exist: we can cancel the 3 above, but this does not imply the existence of .
Thus, we'll talk about isomorphisms - which have two-sided inverses, monomorphisms and epimorphisms - which have cancellability properties, and split morphisms - which are mono's and epi's with correspodning one-sided inverses. We'll talk about what these concepts - defined in terms of equationsolving with arrows - apply to more familiar situations. And we'll talk about how the semantics of some of the more wellknown ideas in mathematics are captured by these notions.
For objects, the properties are interesting in what happens to homsets with the special object as source or target. An empty homset is pretty boring, and a large homset is pretty boring. The real power, we find, is when all homsets with the specific source or target are singleton sets. This allows us to formulate the idea of a 0 in categorical terms, as well as capturing the roles of the empty set and of elements of sets - all using only arrows.
An arrow in a category C is an isomorphism if it has a twosided inverse g. In other words, we require the existence of a such that fg = 1B and gf = 1A.
2.1 In concrete categories
In a category of sets with structure with morphisms given by functions that respect the set structure, isomorphism are bijections respecting the structure. In the category of sets, the isomorphisms are bijections.
2.2 Representative subcategories
Very many mathematical properties and invariants are interesting because they hold for objects regardless of how, exactly, the object is built. As an example, most set theoretical properties are concerned with how large the set is, but not what the elements really are.
If all we care about are our objects up to isomorphisms, and how they relate to each other - we might as well restrict ourselves to one object for each isomorphism class of objects.
Doing this, we get a representative subcategory: a subcategory such that every object of the supercategory is isomorphic to some object in the subcategory.
A groupoid is a category where all morphisms are isomorphisms. The name originates in that a groupoid with one object is a bona fide group; so that groupoids are the closest equivalent, in one sense, of groups as categories.
We say that an arrow f is left cancellable if for any arrows g1,g2 we can show . In other words, it is left cancellable, if we can remove it from the far left of any equation involving arrows.
We call a left cancellable arrow in a category a monomorphism.
3.1 In concrete categories
Left cancellability means that if, when we do first g1 and then f we get the same as when we do first g2 and then f, then we had equality already before we followed with f.
In other words, when we work with functions on sets, f doesn't introduce relations that weren't already there. Anything non-equal before we apply f remains non-equal in the image. This, translated to formulae gives us the well-known form for injectivity:
- or moving out the negations,
Right cancellability, by analogy, is the implication
The name, here comes from that we can remove the right cancellable f from the right of any equation it is involved in.
A right cancellable arrow in a category is an epimorphism.
4.1 In concrete categories
For epimorphims the interpretation in set functions is that whatever f does, it doesn't hide any part of the things g1 and g2 do. So applying f first doesn't influence the total available scope <mamth>g_1</math> and g2 have.
5 Terminal objects
5.1 Pointless sets and global constants
6 Initial objects
7 Zero objects
8 Internal and external hom
- Isomorphisms and existence of inverses.
- Epi- and mono-morphisms and cancellability.
- Examples in concrete categories.
- Monomorphisms and subobjects:
- Factoring through. Equivalence relation by mutual factoring.
- Subobjects as equivalence classes of monomorphisms.
- Splitting and the existence of inverses.
- Terminal and initial objects.
- Constants. Pointless sets.
The arrows of a category are called morphisms. This is derived from homomorphisms.
Some arrows have special properties that make them extra helpful; and we'll name them:
- A morphism with the same object as source and target.
- A morphism that is left-cancellable. Corresponds to injective functions. We say that f is a monomorphism if for any g1,g2, the equation fg1 = fg2 implies g1 = g2. In other words, with a concrete perspective, f doesn't introduce additional relations when applied.
- A morphism that is right-cancellable. Corresponds to surjective functions. We say that f is an epimorphism if for any g1,g2, the equation g1f = g2f implies g1 = g2.
Note, by the way, that cancellability does not imply the existence of an inverse. Epi's and mono's that have inverses realizing their cancellability are called split.
- A morphism is an isomorphism if it has an inverse. Split epi and split mono imply isomorphism. Specifically, is an isomorphism if there is a such that fg = 1w and g = 1v.
- An automorphism is an endomorphism that is an isomorphism.
In a category, we use a different name for the vertices: objects. This comes from the roots in describing concrete categories - thus while objects may be actual mathematical objects, but they may just as well be completely different.
Just as with the morphisms, there are objects special enough to be named. An object v is
- if [v,w] has exactly one element for all other objects w.
- if [w,v] has exactly one element for all other objects w.
- A Zero object
- if it is both initial and terminal.
All initial objects are isomorphic. If i1,i2 are both initial, then there is exactly one map and exactly one map . The two possible compositions are maps and . However, the initiality condition holds even for the morphism set [v,v], so in these, the only existing morphism is and respectively. Hence, the compositions have to be this morphism, which proves the statement.
11 Dual category
The same proof carries over, word by word, to the terminal case. This is an illustration of a very commonly occurring phenomenon - dualization.