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A category consists of three things: a collection of objects, for each pair of objects a collection of morphisms (sometimes call "arrows") from one to another, and a binary operation defined on compatible pairs of morphisms called composition. The category must satisfy an identity axiom and an associative axiom which is analogous to the monoid axioms.

The morphisms must obey the following laws:

1. If u is a morphism from a to b (in short, u:a degreesb), and v:b degreesc, then there is a morphism u degreesv (commonly read "u composed with v") from a to c.

2. Composition of morphisms, where defined, is associative, so if u:a degreesb, v:b degreesc, and w:c degreesd, then (u degreesv) degreesw=u degrees(v degreesw).

3. For each object a, there is an identity morphism I_a, such that for any u:a degreesb, I_a degreesu=u and u degreesI_b=u.

In most concrete categories over sets, an object is some mathematical structure (e.g., a group, vector space, or smooth manifold) and a morphism is a map between two objects. The identity map between any object and itself is then the identity morphism, and the composition of morphisms is just function composition.

One usually requires the morphisms to preserve the mathematical structure of the objects. So if the objects are all groups, a good choice for a morphism would be a group homomorphism. Similarly, for vector spaces, one would choose linear maps, and for differentiable manifolds, one would choose differentiable maps.

In the category of topological spaces, morphisms are usually continuous maps between topological spaces. However, there are also other category structures having topological spaces as objects, but they are not nearly as important as the "standard" category of topological spaces and continuous maps.


See also

Abelian Category, Additive Category, Allegory, Category Theory, Eilenberg-Steenrod Axioms, Groupoid, Holonomy, Logos, Monodromy, Subcategory, Topos Explore this topic in the MathWorld classroom

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References

Freyd, P. J. and Scedrov, A. Categories, Allegories. Amsterdam, Netherlands: North-Holland, 1990.Getzler, E. and Kapranov, M. (Eds.). Higher Category Theory. Providence, RI: Amer. Math. Soc., 1998.Lawvere, F. W. and Schanuel, S. H. Conceptual Mathematics: A First Introduction to Categories. Cambridge, England: Cambridge University Press, 1997.Mac Lane, S. and Gehring, F. W. Categories for the Working Mathematician, 2nd ed. New York: Springer-Verlag, 1998.Munkres, J. R. "Categories and Functors." §28 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 154-160, 1993.

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Cite this as:

Weisstein, Eric W. "Category." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Category.html

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