writing about species based on partiality

SpeciesDiagrams.hs

@@ -400,6 +400,9 @@ objs = enclose 1 1 . vcat' (with & sep .~ 1.5) . (map (\n -> dot # named n))
 data PBij a b where
   PBij :: [a] -> (a -> b) -> PBij a b
 
+applyPBij :: PBij a b -> (a -> b)
+applyPBij (PBij _ f) = f
+
 pbComp :: PBij b c -> PBij a b -> PBij a c
 pbComp (PBij _ f) (PBij as g) = PBij as (f . g)
 

conclusion.lhs

@@ -8,6 +8,9 @@
 \section{Future work}
 \label{sec:future-work}
 
+\todo{Tons of future work.  Feel like I have just scratched the
+  surface---entire research program.}
+
 Theory of molecular and atomic species---translate into HoTT, and see
 precisely what assumptions are needed where.  Will probably shed light
 on the theory as well as have practical applications.
@@ -20,3 +23,7 @@ simply to prevent such infinite families, and hence are not needed in
 such a setting.  It would be interesting to explore minimal criteria
 for analogues of the implicit species theorems in more generalized
 settings
+
+\todo{multisort composition etc.}
+
+\todo{$\BTSub$ and all the attendant stuff!}

defs.lhs

@@ -345,12 +345,12 @@
 \newcategory{Cat}     % (small) categories
 \newcategory{Spe}     % species
 \newcategory{CSpe}    % constructive species
-\newcategory{PSpe}    % partial species
 \newcategory{Hask}    % Haskell types and functions
 \newcategory{B}       % finite sets and bijections
 \newcategory{P}       % naturals and permutations
 \newcategory{L}       % finite linear orders and order-preserving bijections
 \newcommand{\Vect}{\cat{Vec}}    % vector spaces
+\newcommand{\PSpe}{\ensuremath{\prt \Spe}}    % copartial species
 
 \newcommand{\BT}{\mcal{B}}
 \newcommand{\PT}{\mcal{P}}

thesis.bib

@@ -1675,3 +1675,14 @@ @article{lack2014combinatorial
   journal={arXiv preprint arXiv:1402.7151},
   year={2014}
 }
+
+@article{schmitt1993hopf,
+  title={Hopf algebras of combinatorial structures},
+  author={Schmitt, William R.},
+  journal={Canadian journal of mathematics},
+  volume={45},
+  number={2},
+  pages={412--428},
+  year={1993},
+  publisher={University of Toronto}
+}