TITLE: What's more general than category theory?
QUESTION [21 upvotes]: First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then systems are studied that differ from ordinary arithmetic but share some of the same properties (equations where the unknowns represent all sorts of things - even functional equations) and then these properties are abstracted in abstract algebra and whole classes are studied such as groups and rings. Then category theory studies maps between structures (functorial equations), then n-category theory, then ...

Where do we go now? Is category theory the end of the road for the foreseeable future? 
  Is the only way forward to go backwards and generalize in a different direction (like "generalized equations" of optimization or something)?

REPLY [3 votes]: Final Answer: Type Theory and the Univalent Foundations
The Book
The paradox of categorys as a foundational theory is that it aims to divorce itself from set theory but doing so by building on intuitions  and formalisms  that are best explained in a set format.  Thus, CT is the best way to think about sets and to structures that generalize from them and very useful for the study of set-like structures.  But the further we go away from the traditional attributes of a set, the more category theory becomes contentious or at least more difficult to work with. Normally, this is "swept under the rug" by instead of focusing on objects and their morphisms we focus on classes and their morphisms, in a sense taking us away from the strictly quantitative nature of set theory into the more qualitative world of classes. 
But in my opinion, this is like drawing a bunch of one's and zero's on paper with charcoal, then smudging it with an eraser to look like a non-descript face, and then saying that you have progressed from the class of  set (you can count the number of smudges) to a class of faces (you can't count the faces the smudge represents) when in fact  the face that you can represent will always be limited by the initial scoring of the one and zero in charcoal and thus still reflect a structurally set-based nature
My opinions on the implicit presence of set theory within categories aside,  the notion that a type of object and then what you can do on that type  is a construct that is familiar to every programmer in the world. So to are the notions of polymorphism  and inheritance childs play to an object oriented programmer yet they are very difficult to deal with in a basic categorical perspective since they require one object to have multiple identities and this leads to construct such as multi-categories, or colored categories, etc. But since type theory is  built from day one as it where on the more abstract notion of a type (which for all intents and purposes can be considered the categorical class), then it is better equipped  to deal with structures that bear absolutely no relation to sets.  
In effect, type theory allows you to draw the face exactly and then deal with that construct as its own mathematical identity, be it's own "thing" instead of having to  build up that identity digitally and be limited by the combinations of smudged ones and zeros.