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In category theory, a branch of mathematics, the **diagonal functor** is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects *within* the category : a product is a universal arrow from to . The arrow comprises the projection maps.

More generally, given a small index category , one may construct the functor category , the objects of which are called diagrams. For each object in , there is a constant diagram that maps every object in to and every morphism in to . The diagonal functor assigns to each object of the diagram , and to each morphism in the natural transformation in (given for every object of by ). Thus, for example, in the case that is a discrete category with two objects, the diagonal functor is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram , a natural transformation (for some object of ) is called a cone for . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category , and a limit of is a terminal object in , i.e., a universal arrow . Dually, a colimit of is an initial object in the comma category , i.e., a universal arrow .

If every functor from to has a limit (which will be the case if is complete), then the operation of taking limits is itself a functor from to . The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor.

For example, the diagonal functor described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. Other well-known examples include the pushout, which is the limit of the span, and the terminal object, which is the limit of the empty category.

In category theory, a branch of mathematics, when we say that a construction satisfies a **universal property**, it means that this construction can be seen as an initial or terminal object of some other category. By "universal property" one may mean either a universal initial or terminal morphism. The nomination of initial/terminal morphisms comes from the fact that these two morphisms are respectively initial/terminal objects in their corresponding comma categories which has been made from the original ones. Intuitively, by *universal* we mean that the initial/terminal morphism(s) is (are) the "most general" constructions in that category with those properties.

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 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, a **pre-abelian category** is an additive category that has all kernels and cokernels.

In mathematics, a **direct limit** is a way to construct a object from many objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by .

In category theory, the **product** of two 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.

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 mathematics, the **gluing axiom** is introduced to define what a sheaf *F* on a topological space *X* must satisfy, given that it is a presheaf, which is by definition a contravariant functor

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

**Kan extensions** are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.

In category theory, a **strict 2-category** is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over **Cat**.

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two *coherence maps*—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the **diagonal morphism**

In category theory, a branch of mathematics, a **presheaf** on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.

In category theory, a branch of mathematics, **dagger compact categories** first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations. They also appeared in the work of John Baez and James Dolan as an instance of semistrict *k*-tuply monoidal *n*-categories, which describe general topological quantum field theories, for *n* = 1 and *k* = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.

In category theory, a branch of mathematics, the **cone of a functor** is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

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, especially in algebraic topology, the **homotopy limit and colimit** are variants of the notions of limit and colimit. They are denoted by holim and hocolim, respectively.

In category theory, a branch of mathematics, the **density theorem** states that every presheaf of sets is a colimit of representable presheaves in a canonical way.

- Mac Lane, Saunders; Moerdijk, Ieke (1992).
*Sheaves in geometry and logic a first introduction to topos theory*. New York: Springer-Verlag. pp. 20–23. ISBN 9780387977102.

- May, J. P. (1999).
*A Concise Course in Algebraic Topology*(PDF). University of Chicago Press. p. 16. ISBN 0-226-51183-9.

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