In category theory, the **product** of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Fix a category **C**. Let *X*_{1} and *X*_{2} be objects of **C**. A product of *X*_{1} and *X*_{2} is an object *X*, typically denoted *X*_{1} × *X*_{2}, equipped with a pair of morphisms *π*_{1} : *X* → *X*_{1}, *π*_{2} : *X* → *X*_{2} satisfying the following universal property:

- For every object
*Y*and every pair of morphisms*f*_{1}:*Y*→*X*_{1},*f*_{2}:*Y*→*X*_{2}, there exists a unique morphism*f*:*Y*→*X*_{1}×*X*_{2}such that the following diagram commutes:

Whether a product exists may depend on **C** or on *X*_{1} and *X*_{2}. If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of *the* product.

The morphisms *π*_{1} and *π*_{2} are called the ** canonical projections ** or **projection morphisms**. Given *Y* and *f*_{1}, *f*_{2}, the unique morphism *f* is called the **product of morphisms***f*_{1} and *f*_{2} and is denoted ⟨*f*_{1}, *f*_{2}⟩.

Instead of two objects, we can start with an arbitrary family of objects indexed by a set *I*.

Given a family (*X*_{i})_{i∈I} of objects, a **product** of the family is an object *X* equipped with morphisms *π*_{i} : *X* → *X*_{i} satisfying the following universal property:

- For every object
*Y*and every*I*-indexed family of morphisms*f*_{i}:*Y*→*X*_{i}, there exists a unique morphism*f*:*Y*→*X*such that the following diagrams commute for all*i*in*I*:

The product is denoted ∏_{i∈I}*X*_{i}. If *I* = {1, …, *n*}, then it is denoted *X*_{1} × ⋯ × *X*_{n} and the product of morphisms is denoted ⟨*f*_{1}, …, *f*_{n}⟩.

Alternatively, the product may be defined through equations. So, for example, for the binary product:

- Existence of
*f*is guaranteed by existence of the operation ⟨⋅, ⋅⟩. - Commutativity of the diagrams above is guaranteed by the equality ∀
*f*_{1}, ∀*f*_{2}∀*i*∈ {1, 2},*π*_{i}∘ ⟨*f*_{1},*f*_{2}⟩ =*f*_{i}. - Uniqueness of
*f*is guaranteed by the equality ∀*g*:*Y*→*X*_{1}×*X*_{2}, ⟨*π*_{1}∘*g*,*π*_{2}∘*g*⟩ =*g*.^{ [1] }

The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set *I* considered as a discrete category. The definition of the product then coincides with the definition of the limit, { *f* }_{i} being a cone and projections being the limit (limiting cone).

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take **J** as the discrete category with two objects, so that **C**^{J} is simply the product category **C** × **C**. The diagonal functor Δ : **C** → **C** × **C** assigns to each object *X* the ordered pair (*X*, *X*) and to each morphism *f* the pair (*f*, *f*). The product *X*_{1} × *X*_{2} in **C** is given by a universal morphism from the functor Δ to the object (*X*_{1}, *X*_{2}) in **C** × **C**. This universal morphism consists of an object *X* of **C** and a morphism (*X*, *X*) → (*X*_{1}, *X*_{2}) which contains projections.

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets *X*_{i} the product is defined as

- ∏
_{i∈I}*X*_{i}:= { (*x*_{i})_{i∈I}| ∀*i*∈*I*,*x*_{i}∈*X*_{i}}

with the canonical projections

*π*_{j}: ∏_{i∈I}*X*_{i}→*X*_{j},*π*_{j}((*x*_{i})_{i∈I}) :=*x*_{j}.

Given any set *Y* with a family of functions *f*_{i} : *Y* → *X*_{i}, the universal arrow *f* : *Y* → ∏_{i∈I}*X*_{i} is defined by *f*(*y*) := (*f*_{i}(*y*))_{i∈I}.

Other examples:

- In the category of topological spaces, the product is the space whose underlying set is the Cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous.
- In the category of modules over some ring
*R*, the product is the Cartesian product with addition defined componentwise and distributive multiplication. - In the category of groups, the product is the direct product of groups given by the Cartesian product with multiplication defined componentwise.
- In the category of graphs, the product is the tensor product of graphs.
- In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.)
- In the category of algebraic varieties, the product is given by the Segre embedding.
- In the category of semi-abelian monoids, the product is given by the history monoid.
- A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).

An example in which the product does not exist: In the category of fields, the product **Q** × **F**_{p} does not exist, since there is no field with homomorphisms to both **Q** and **F**_{p}.

Another example: An empty product (i.e. *I* is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group *G* there are infinitely many morphisms , so *G* cannot be terminal.

If *I* is a set such that all products for families indexed with *I* exist, then one can treat each product as a functor **C**^{I} → **C**.^{ [2] } How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For *f*_{1} : *X*_{1} → *Y*_{1}, *f*_{2} : *X*_{2} → *Y*_{2} we should find a morphism *X*_{1} × *X*_{2} → *Y*_{1} × *Y*_{2}. We choose ⟨ *f*_{1} ∘ π_{1}, *f*_{2} ∘ π_{2} ⟩. This operation on morphisms is called **Cartesian product of morphisms**.^{ [3] } Second, consider the general product functor. For families {*X*}_{i},{*Y*}_{i}, *f*_{i} : *X*_{i} → *Y*_{i} we should find a morphism ∏_{i∈I}*X*_{i} → ∏_{i∈I}*Y*_{i}. We choose the product of morphisms {*f*_{i} ∘ π_{i}}_{i}.

A category where every finite set of objects has a product is sometimes called a **Cartesian category**^{ [3] } (although some authors use this phrase to mean "a category with all finite limits").

The product is associative. Suppose **C** is a Cartesian category, product functors have been chosen as above, and 1 denotes a terminal object of **C**. We then have natural isomorphisms

These properties are formally similar to those of a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category.

For any objects X, Y, and Z of a category with finite products and coproducts, there is a canonical morphism *X* × *Y* + *X* × *Z* → *X* × (*Y* + *Z*), where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct *X* × *Y* + *X* × *Z* guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):

The universal property of the product *X* × (*Y* + *Z*) then guarantees a unique morphism *X* × *Y* + *X* × *Z* → *X* × (*Y* + *Z*) induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism

- Coproduct – the dual of the product
- Diagonal functor – the left adjoint of the product functor.
- Limit and colimits
- Equalizer
- Inverse limit
- Cartesian closed category
- Categorical pullback

In mathematics, specifically category theory, a **functor** is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In category theory, a branch of mathematics, a **universal property** is an important property which is satisfied by a **universal morphism**. Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category. Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

In category theory, a branch of mathematics, the abstract notion of a **limit** captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a **colimit** generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In category theory, a branch of mathematics, an **initial object** of a category C is an object I in C such that for every object X in C, there exists precisely one morphism *I* → *X*.

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In category theory, a **category is Cartesian closed** if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In category theory, the **coproduct**, or **categorical sum**, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, the category **Ab** has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in **Ab**.

In category theory, a branch of mathematics, a **pushout** is the colimit of a diagram consisting of two morphisms *f* : *Z* → *X* and *g* : *Z* → *Y* with a common domain. The pushout consists of an object *P* along with two morphisms *X* → *P* and *Y* → *P* that complete a commutative square with the two given morphisms *f* and *g*. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .

In mathematics, a **comma category** is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category".

In category theory, a branch of mathematics, a **pullback** is the limit of a diagram consisting of two morphisms *f* : *X* → *Z* and *g* : *Y* → *Z* with a common codomain. The pullback is often written

**Fibred categories** are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which *inverse images* of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space *X* to another topological space *Y* is associated the pullback functor taking bundles on *Y* to bundles on *X*. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

In mathematics, specifically in category theory, *F*-**algebras** generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor *F*, the *signature*.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called **hom-functors** and have numerous applications in category theory and other branches of mathematics.

In category theory, a branch of mathematics, a **diagram** is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a *function* from a fixed index *set* to the class of *sets*. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a *functor* from a fixed index *category* to some *category*.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In algebraic geometry, a **prestack***F* over a category *C* equipped with some Grothendieck topology is a category together with a functor *p*: *F* → *C* satisfying a certain lifting condition and such that locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.

- ↑ Lambek J., Scott P. J. (1988).
*Introduction to Higher-Order Categorical Logic*. Cambridge University Press. p. 304. - ↑ Lane, S. Mac (1988).
*Categories for the working mathematician*(1st ed.). New York: Springer-Verlag. p. 37. ISBN 0-387-90035-7. - 1 2 Michael Barr, Charles Wells (1999).
*Category Theory – Lecture Notes for ESSLLI*. p. 62. Archived from the original on 2011-04-13.

- Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).
*Abstract and Concrete Categories*(PDF). John Wiley & Sons. ISBN 0-471-60922-6. - Barr, Michael; Charles Wells (1999).
*Category Theory for Computing Science*(PDF). Les Publications CRM Montreal (publication PM023). Archived from the original (PDF) on 2016-03-04. Retrieved 2016-03-21. Chapter 5. - Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics**5**(2nd ed.). Springer. ISBN 0-387-98403-8. - Definition 2.1.1 in Borceux, Francis (1994).
*Handbook of categorical algebra*. Encyclopedia of mathematics and its applications 50–51, 53 [i.e. 52]. Volume 1. Cambridge University Press. p. 39. ISBN 0-521-44178-1.`|volume=`

has extra text (help)

- Interactive Web page which generates examples of products in the category of finite sets. Written by Jocelyn Paine.
- Product in
*nLab*

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