A closed category, in the right monoidal sense, is a monoidal category in which tensoring on the right with any object 
has a right adjoint
functor, written ![[X,-]](/images/equations/ClosedCategory/Inline2.svg)
.
Consequently, morphisms 
correspond naturally to morphisms ![A->[X,Y]](/images/equations/ClosedCategory/Inline4.svg)
. Eilenberg and Kelly (1966)
also define a more general notion of closed category in which hom objects
and unit structure are specified directly, without assuming a monoidal
category structure.
Closed Category
See also
Category, Functor, Monoidal Category, Tensor CategoryExplore with Wolfram|Alpha
References
Eilenberg, S. and Kelly, G. M. "Closed Categories." In Proceedings of the Conference on Categorical Algebra: La Jolla 1965 (Ed. S. Eilenberg, D. K. Harrison, S. MacLane, and H. Röhrl). New York: Springer-Verlag, pp. 421-562, 1966. https://doi.org/10.1007/978-3-642-99902-4_22.Cite this as:
Weisstein, Eric W. "Closed Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedCategory.html