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A category $\mathcal{C}$ consists of the following data:
1. 2. for each ordered pair $(A,B)$ of objects of $\mathcal{C}$, a collection (we will assume it is a set) $\hom(A,B)$ of morphisms from the domain $A$ to the codomain $B$
3. a function $\circ:\hom(A,B)\times\hom(B,C)\to\hom(A,C)$ called composition.
We normally denote $\circ(f,g)$ by $g\circ f$ for morphisms $f,g$. The above data must satisfy the following axioms: for objects $A,B,C,D$,
A1: $\hom(A,B)\cap\hom(C,D)=\emptyset$ whenever $(A,B)\neq(C,D)$, i.e. the intersection is nontrivial only when $A=C$ and $B=D$.
A2: (Associativity) if $f\in\hom(A,B)$, $g\in\hom(B,C)$ and $h\in\hom(C,D)$, $h\circ(g\circ f)=(h\circ g)\circ f$
A3: (Existence of an identity morphism) for each object $A$ there exists an identity morphism ${}id_{{A}}\in\hom(A,A)$ such that for every $f\in\hom(A,B)$, $f\circ id_{{A}}=f$ and ${}id_{{A}}\circ g=g$ for every $g\in\hom(B,A)$.
Some examples of categories:

0 is the empty category with no objects or morphisms, 1 is the category with one object and one (identity) morphism.

If we assume we have a universe $U$ which contains all sets encountered in βeverydayβ mathematics, Set is the category of all such small sets with morphisms being set functions

Top is the category of all small topological spaces with morphisms continuous functions

Grp is the category of all small groups whose morphisms are group homomorphisms
Remark. If $\hom(A,B)$ in the second condition above is not required to be a set (but a class), we usually call $\mathcal{C}$ a large category.
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Attached Articles
category associated to a partial order by archibal
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algebra formed from a category by rspuzio
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Corrections
Also defines: object by kompik β
fix linking policy by Mathprof β
italicized ob and linking by Wkbj79 β
large category by CWoo β
Comments
"small"
What is meant by the qualifier "small" in the examples? That is, when is a group "small" in this sense? A topological space? Does this just mean that their underlying object must be a set? (I had assumed that was part of their definition, but I suppose it need not be... for a topological space, this would seem to pose problems...)
Andrew
Re: "small"
The term ``small'' is relativean object is small if and only if it is a member of the universe under consideration. It is basically a way of giving a nod to the restricted comprehension principle. You may find the discussions in S. MacLane, _Categories for the Working Mathematician_ (2nd ed., chapter 1) and F. W. Lawvere and R. Rosebrugh, _Sets for Mathematics_ (chapter 6 and appendix C) interesting.
Re: "small"
It might be worth mentioning this in the entry, then. The only usage of "small" I have nocuntered in this context is rather different: the objects in a category were not required to form a set (although Hom(A,B) was required to be a set for every A and B). A small category, then, was one in which the isomorphism classes of objects formed a set.
In a sense, this is the same usage: a small category is one in which one does not have settheoretic problems with various constructions (enough injectives, for example).
What is an object?
Supposedly, the definition of "category" also defines "object". Reading through the definition, it is not clear to me how this is done, or what the definition of "object" is. All I know for certain is that a class of them is one of the things that a category consists of, and that morphisms are defined in terms of pairs of them.
Re: What is an object?
An object is an element of the class of objects of a category. You can't really say anything more than that, since an object could be anything.
Re: What is an object?
To paraphrase what Hilbert said about points and lines in geometry,
they could be chairs and tables or anything else. If something
quacks like an object and walks like an object, then it's an object.
In the modern abstract axiomatic approach to mathematics, we have
various axiomatic systems such as category theory, group theory,
Euclidean geometry, nonEuclidean geometry, real numbers, topology,
measure theory, etc. The way these systems work is that one has
a certain number of primitive terms ("morphism" and "object" in the
case of category theory, "point" and "line" in geometry) and certain axioms, which are statements about these objects. (One can use the
term "defining property" as a synonym for "axiom").
In this approach, the primitive terms are not defined in the
conventional sense of the term. The reason for this is that,
in order to define something, we need to define it in terms of
something else. But then, to define that something else, we
would need to define it in terms of yet something else. And so
on and so forth. (If you are familiar with it, the story of
Bertrand Russell and "turtles all the way down" is quite an apt
metaphor here in more way than one.) To avoid such a neverending
descent of definitions which would never let us get started, what
we instead do is to stop at some point, say "the buck stops here"
and agree not to define certain terms.
Rather, what we do is to say that these terms can refer to anything
as long as this is done in such a way that the axioms are satisfied.
The point of Hilbert's quip was that, if, say, the arrangement of
tables and chairs in some restaurant is such that the axioms of geometry are a satisfied (say, every two tables share a chair
in common and, given two chairs, there is only one table which
has both of these chairs) then we are perfectly well justified in
referring the the chairs as "points" and to the tables as "lines"
and considering the eatery as some sort of geometrical space,
however odd this usage might seem. Likewise, if we were studying
the genealogy of ducks, then we could use the language of category
theory by terming our ducks "objects" and terming the the relation
of one duck as ancestor to another duck (with the curious convention
that we speak of a duck as its own ancestor in a trivial sense) as
"morphisms". Returning to more conventional examples, it is
permissible to call geodesics on a tractrix "lines" in the
context of Beltrami's model of nonEuclidean geometry or to term
vectors (of possibly different sizes) as "objects" and term matrices
(possible rectangular, not just square) "morphisms" or to use the
term "object" to refer to a topological space and the term "morphism"
to refer to a continuous map.
This attitude towards definitions is actually quite close to how
we use language in general. Think of a dictionary. In Webster's
tome one finds definitions of all the words of the English language.
However, these words are defined using the very same English
language which th work purports to define. This may
sound suspiciously circular  how can you define something in
terms of itself !!?!????!? Well, looking at the matter more
closely, we see that this isn't quite the case. Rather, what the
dictionary really does is to describe the relations between words.
Likewise, there is the possibility of different usages. For
instance, when I say "keyboard", I mean a certain set of keys on
which I am typing this note, but Joe might mean the keys on his
typewriter, and at a later time, when taking a break to play some
music, I might be referring to the user interface of my harpsichord.
All these are legitimate uses of the terms "key" and "keyboard"
in their respective contexts since they all conform to the
definitions in the dictionary, which would go something like "A
key is thing on a keyboard which one presses in order to perform a
particular action" and "A keyboard is a collection of keys which
serve as a tactile interface to a machine". Note that these
definitions define "key" and "keyboard" in terms of each other
in much the same way that the axioms of projective geometry define
"line" and "point" in terms of each other or the axioms of
category theory define "object" and "morphism" in terms of each other.
Likewise, in both everyday life, and mathematics, context is crucial
to the use of language. When we hear someone use a term, we need
to understand the context, otherwise one can spout nonsense like
thinking that I am replying to your post on a harpsichord! When
reading a piece of mathematical writing which uses the language of
category theory, we need to understand how the terms "morphism" and
"object" are being employed  is the author discussing a particular
category in which case they have a more specific meaning or talking
about these concepts in general as in a book on category theory or
what? While we can rest assured that, if the author did not make a
mistake (in which case we should file a correction :) ) that these
words are being used in a way consistent with the defining axioms of
category theory, we need to examine the context to understand what
is being said fully.
Re: What is an object?
All well and good. However, it would be useful to indicate which terms are primitive.
Re: What is an object?
> However, it would be useful to indicate which terms are primitive.
They are so indicated in the entry  items 1. 2. and 3. at the top
list the primitive terms, after which the axioms are given.
Re: What is an object?
Another potentially stupid question: I have two corrections outstanding, I know one of them is for the happy ending problem. How do I find out which one the other one is for without having to look at 14 pages of objects?
Lisa
"And in 5 years, you're gonna be blown away!"  Jennifer Granholm
Finding corrections (was: What is an object?)
CompositeFan writes:
> I have two corrections outstanding, I know one
> of them is for the happy ending problem. How do
> I find out which one the other one is for without
> having to look at 14 pages of objects?
There's no way to see just your outstanding corrections. I remember that someone pointed out this problem before.
However, you can greatly reduce the number of pages by choosing 150 items per page in your preferences. (For some reason it will only show 75 corrections per page, rather than 150, but in your case that still means only two pages.)
Re: What is an object?
The other one is for "proof of Catalan's Identity".
Re: Finding corrections (was: What is an object?)
That's a great idea, I'll do that! Thanks.
link to 'class' is incorrect
The word "class" links to "Vizing's theorem", instead of "Class": http://planetmath.org/encyclopedia/Class.html
inverses of morphisms
What can be said if there is an inverse of a morphism f. Ie, there is a g (obviously in Mor(Y,X)) st g o f = Ix and f o g = Iy. Is this inverse unique? if so how can one prove it?
Re: inverses of morphisms
They are indeed unique.
Suppose g,h\in Mor(Y,X) satisfy
gf=fg=1
hf=fh=1.
How can we get g=h?
Cam
Re: inverses of morphisms
gf=fg=1
hf=fh=1.
oooh, ok, would it be ?
gf = 1
gfh = 1h
gfh = h
g = h (as fh = 1)
Re: inverses of morphisms
Looks good to me!
Cam