Research Program
Twosome
 Twosome
 For existence rather than participation
 Examples:
Opposites coexist and all is the same
 Tensor product (opposites coexist, in parallel, geometry (homogeneous choice)) vs. Homset functor (all is the same, in series, algebra (stepbystep)).
Free will and fate
 Free will (range) and fate (domain)  functions  everything in the domain must get assigned an output but not everything in the range must get assigned an input.

 Limits (softwiring external relationships) colimits (hardwiring internal structure). Softwiring precedes hardwiring.
Outside and inside
 Outsideinside (orientable  nonorientable)
"Interestingly, a graphic has the power to evoke feelings of understanding, without really meaning much. The same is true for text: it is possible to use a language such as English to express ideas that are never made rigorous or clear. When someone says “I believe in free will,” what does she believe in? We may all have some concept of what she’s saying—something we can conceptually work with and discuss or argue about. But to what extent are we all discussing the same thing, the thing she intended to convey?" Spivak, Category Theory for Scientists (free version) pg.7
Theory and practice
 Algorithm off and algorithm on
Same and different
 Same (indistinguishable)  different (distinguishable). Twelvefold way  combinatorics.
 Sameness + difference. (Dvejybės atvaizdas) (Same means "combine like units" and different means "list separate units")
 There are only two (1)categories, "same" ({$x=y$}) and "different" ({$x\neq y$}), for they are the possible values of {$\textrm{hom}(x,y)$}, which are the 0morphisms, which are simply the identity morphisms in a set of objects. If {$y=x$}, then we have the identity morphism, and otherwise we have the empty set.
Abstract theory (questions) and concrete examples (answers)
Type and term
 Type is a question, term is an answer.
 Type is a unit, term is an amount.
 Logically, the point of any term is an answer.
 Not having a term is not having an answer. An answer "no" is rather an answer to the negation of the type, an answer about the negation. This is the essence of type theory.
Nonexamples
 Discuss yinyang.
 Seven kinds of duality  sevensome. Will relate to Bott periodicity.
 Malefemale gender.
 Twocycle Bott periodicity for complex numbers.
 nLab: Aufhebung: The mathematics of yin and yang Lawvere's example of {$N_{even}$} and {$N_{odd}$} illustrates same and different.