some element of the set: the "basepoint". Morphisms are functions on sets which map basepoints to basepoints. In a similar fashion one can form the category of pointed spaces . The category of associative algebras over a ring is the coslice category , since any ring homomorphism induces an associative -algebra structure on , and vice versa. Morphisms are then maps that make the diagram commute. The category of graphs is , with the functor taking a set to . The objects then consist of two sets and a function; is an indexing set, is a set of nodes, and chooses pairs of elements of for each input from . That is, picks out certain edges from the set of possible edges. A morphism in this category is made up of two functions, one on the indexing set and one on the node set. They must "agree" according to the general definition above, meaning that must satisfy . In other words, the edge corresponding to a certain element of the indexing set, when translated, must be the same as the edge for the translated index. Many "augmentation" or "labelling" operations can be expressed in terms of comma categories. Let be the functor taking each graph to the set of its edges, and let be (a functor selecting) some particular set: then is the category of graphs whose edges are labelled by elements of . This form of comma category is often called objects -over - closely related to the "objects over " discussed above. Here, each object takes the form , where is a graph and a function from the edges of to . The nodes of the graph could be labelled in essentially the same way. A category is said to be locally cartesian closed if every slice of it is cartesian closed (see above for the notion of slice). Locally cartesian closed categories are the classifying categories of dependent type theories. Limits and universal morphisms Limits and colimits in comma categories may be "inherited". If and are complete, is a continuous functor, and is another functor (not necessarily continuous), then the comma category produced is complete, and the projection functors and are continuous. Similarly, if and are cocomplete, and is cocontinuous, then is cocomplete, and the projection functors are cocontinuous. For example, note that in the above construction of the category of graphs as a comma category, the