nLab
direct product

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Category theory

category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Limits and colimits

limits and colimits

1-Categorical

2-Categorical

(∞,1)-Categorical

Model-categorical

Direct products

Definition

In universal algebra, a direct product is simply a product in a concrete category that is created by the forgetful functor.

Compare the direct sum, a more complicated concept.

Examples

Trivially, a cartesian product of sets is a direct product in Set.

One of the requirements of a topological category is that any family of objects must have a direct product, although the term ‘direct product’ is not used in topology.

Many algebraic categories, such as Grp, Ab, Ring, etc, also have all direct products; this is where the term ‘direct product’ originated.

The category of (Set-valued) models of any Lawvere theory has all direct products; this includes the examples from algebra above.

Revised on August 26, 2012 23:54:41 by Urs Schreiber (89.204.139.66)
Edit | Back in time (2 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Ab, end, Kan extension, limit, cone, pullback, initial object, copower, image, product, disjoint coproduct, coproduct, colimit, pushout, intersection, homotopy limit, 2-limit, comma category, homotopy pullback, limit in a quasi-category, disjoint union, quotient object, wedge sum, sketch, wide pullback, equalizer, Yoneda extension, exact functor, direct sum, direct product, mapping cone, base change, 2009 March changes, final functor, ind-object in an (infinity,1)-category, dependent product, coimage, 2009 May changes, fiber sequence, chain homology and cohomology, fiber, hypercomplex number, ultraproduct, filtered limit, infinity-limits - contents, reflexive coequalizer, pro-object in an (infinity,1)-category, preserved limit, final (infinity,1)-functor, (infinity,1)-Kan extension, created limit, connected limit, DHR category, finite limit, sifted colimit, colimits in categories of algebras, projection, cross product, product topological space, contents of contents, measurable locale, Coleman-Mandula theorem, flexible limit, small limit, direct sum of Banach spaces, exactness property, Baer sum, bilinear map, cofiber, dependent sum, coherence theorem for bicategories with finite limits, strict initial object, cofiber sequence, van Kampen colimit