Cartesian-closed category

A category $\mathfrak{C}$ such that the following axioms are satisfied:

A1) there exists a terminal object $\mathbf{1}$;

A2) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist a product $A \times B$ and given projections $\mathrm{pr}_1 : A \times B \rightarrow A$, $\mathrm{pr}_2 : A \times B \rightarrow B$;

A3) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist an object $A^B$ and an evaluation arrow $\mathrm{ev} : A^B \times B \rightarrow A$ such that for any arrow $F : C \times A \rightarrow B$ there is a unique arrow $[f] : C \rightarrow A^B$ with $\mathrm{ev}\circ ([f]\times \mathrm{id}_A) = f$.

These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors $$ \mathfrak{C} \rightarrow \mathbf{1}\,\ \ c \mapsto 0\,; $$ $$ \mathfrak{C} \rightarrow \mathfrak{C} \times \mathfrak{C}\,\ \ c \mapsto \langle c,c \rangle \,; $$ $$ \mathfrak{C} \rightarrow \mathfrak{C}\,\ \ c \mapsto c \times b $$ have each a specified right-adjoint, written respectively as: $$ 0 \mapsto t\,; $$ $$ \langle a,b \rangle \mapsto a \times b\,; $$ $$ c \mapsto c^b \ . $$

Some examples of Cartesian-closed categories are:

E1) any Heyting algebra $\mathcal{H}$;

E2) the category $\mathsf{Set}^{\mathfrak{C}}$ for any small category $\mathfrak{C}$ with $\mathsf{Set}$ the category of (small) sets — in particular $\mathsf{Set}$ itself;

E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos;

E4) any elementary topos $\mathcal{E}$;

E5) the category $\mathsf{Cat}$ of all (small) categories;

E6) the category $\mathsf{Graph}$ of graphs and their homomorphisms;

E7) the category $\omega$-$\mathsf{CPO}$ of $\omega$-CPOs.

These definitions can all be put into a purely equational form.

References

[a1]	M. Barr, C. Wells, "Category theory for computing science" , CRM (1990)
[a2]	J. Lambek, P.J. Scott, "Introduction to higher order categorical logic" , Cambridge Univ. Press (1986)
[a3]	S. MacLane, I. Moerdijk, "Sheaves in geometry and logic" , Springer (1992)
[a4]	S. MacLane, "Categories for the working mathematician" , Springer (1971)