2. Literature Review
2.1. Kuratowski Closure Operator
A closure operator is an arbitrary set-valued, set-function 
where 
is the power set of a non-void set 
that satisfies some closure axioms [1]. Consequently, various combinations of the following axioms have been used in the past in an attempt to define closure operators [2]. Let
.
1) Grounded: 
2) Expansive: 
3) Sub-additive:
. This axiom implies the Isotony axiom:
implies 
4) Idempotent: 
The structure
, where 
satisfies the first three axioms is called a closure space [2].
2.2. Isotonic Space
A closure space 
satisfying only the grounded and the Isotony closure axioms is called an isotonic space [3]. This is the space of interest in this study and clearly, it is more general than a closure space.
In a dual formulation, a space 
is isotonic if and only if the interior function
satisfies;
1)
.
2) 
implies 
2.3. Category
A category has objects 
and arrows 
such that
, i.e. 
and
. Two arrows 
and 
such that 
are said to be composable [4].
Axioms of a Category
According to [5], the following are the axioms of a category;
1) If 
and 
are composable, then they must have a composite which is the arrow shown 
shown in the diagram below
The arrow 
goes from the 
to the 
such that 
and the
1) For every object 
there exists the identity arrow
.
2) Composition is associative. This can be represented in as shown below;
3. Main Results
3.1. Quasi-Isotone Space
A closure space 
with a closure operator 
is called a quasi-isotone space if the closure operator satisfies the following three Kuratowski closure axioms 1) 
2) For 
implies 
3)
.
The third axiom is called the idempotent axiom. It will become very useful while defining the pseudo-category on the quasi-isotone space.
3.2. Pseudo-Category
To define a pseudo-category on the class of quasi-isotone space, we firstly need to identify the objects and morphisms on this class of spaces. The objects are the closure operators 
such that they obey the three Kuratowski axioms above.
Next is to define the morphisms on the category. The arrows linking the objects together are 
such that
. More explicitly, the arrow 
may be represented diagrammatically by;
Therefore, the pseudo-category on quasi-isotone space has as objects the closure operators 
and 
such that 
as the morphisms. Of course two arrows 
and 
such that 
are said to be composable
Axioms of the Pseudo-Category
1) If 
and 
are composable, then they must have a composite which is the arrow 
shown in the diagram below
The arrow 
goes from the 
to the 
such that 
and the
.
2) For every object 
there exists the identity arrow
. The existence of this identity arrow is guaranteed by the idempotent axiom defined on the quasi-isotone axiom. Indeed, the name pseudo-category for this structure is adopted since the idempotent axiom is not exactly an identity function.
3) Composition is associative. This can be representedas in the diagram below:
4. Remark
Other notions of a category may also be defined on the pseudo-category of quasi-isotone spaces. They include functors, natural transformations, adjunctions among others.
5. Conclusion
On a space defined by the Kuratowski closure axioms, it is possible to define a category-like structure in a very natural and straightforward way. This will enable some mathematical analysis to be extended onto closure spaces.
[1] W. J. Thron, “What Results Are Valid on Cech-Closure Spaces,” Topology Proceedings, Vol. 6, No. 3, 1981, pp. 135-158.
[2] T. A Sunitha, “A Study of Cech Closure Spaces,” Doctor of Philosophy Thesis, School of Mathematical Sciences, Cochin University of Science and Technology, Cochin, 1994.
[3] A. K. Elzenati and E. D. Habil, “Connectedness in Isotonic Spaces,” Turkish Journal of Mathematics, Vol. 30, No. 3, 2006, pp. 247-262.
[4] C. McLarty, “Elementary Categories, Elementary Toposes,” Oxford University Press, Oxford, 1992.
[5] S. MacLane, “Category for the Working Mathematician,” 2nd Edition, Springer-Verlag Inc., New York, 1998.