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For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
Universal properties come in two types: initial properties and terminal properties. Since these are dual notions, it is only necessary to discuss one of them. The idea of using an initial property is to set up the problem in terms of some auxiliary category E , so that the problem at hand corresponds to finding an initial object of E .
Another example: An empty product (that is, 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 there are infinitely many morphisms , so cannot be terminal.
Then the above free–forgetful adjunction involving the Eilenberg–Moore category is a terminal object in (,). An initial object is the Kleisli category , which is by definition the full subcategory of C T {\displaystyle C^{T}} consisting only of free T -algebras, i.e., T -algebras of the form T ( x ) {\displaystyle T(x)} for some object x of C .
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 J {\displaystyle {\mathcal {J}}} is a discrete category with two objects, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow ...
Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc. If F : C → D is an equivalence of categories, and G 1 and G 2 are two inverses of F, then G 1 and G 2 are naturally isomorphic.
terminal 1. An object A is terminal (also called final) if there is exactly one morphism from each object to A; e.g., singletons in Set. It is the dual of an initial object. 2. An object A in an ∞-category C is terminal if (,) is contractible for every object B in C. thick subcategory