I am a retired mathematician and mathematical logician (BS, MIT '65, and PhD. Boston U. '71) associated as a Visiting Scholar at the University of California at Riverside (John Baez's home base too). My mentor in mathematics was Gian-Carlo Rota of MIT who died prematurely in 1999 (age 66). I then focused on one of his unfinished projects which was to develop a logic of equivalence relations = partitions = quotient sets, which would be a type of dual to the usual Boolean logic of subsets (usually mis-specified as the special case of propositional logic) since quotient sets are categorically dual to subsets. Dedekind had defined the join and meet of partitions in the 19th century, but one needs at least an implication operation on partitions to warrant calling it a "logic of partitions." After figuring out how to define the implication and further seeing the algorithm to translate all the Boolean operations on subsets over to partitions, the long paper developing partition logic (along with correctness and completeness theorems for a system of semantic tableaus for partitions) was published in the [Review of Symbolic Logic](http://www.ellerman.org/the-logic-of-partitions/). I gave the definition of implication for partitions in a comment on Lesson 11. A shorter and better introduction to partition logic was later published in the [Logic Journal of the IGPL](http://www.ellerman.org/introduction-to-partition-logic/).

The quantitative version of Boole's logic of subsets is just finite logical probability theory as developed by Boole in his _Laws of Thought_. Rota always held that probability (i.e., the normalized size of a subset) is to subsets as information is to partitions.

$$\frac{\text{Probability}}{\text{Subsets}}=\frac{\text{Information}}{\text{Partitions}}$$

The key to working that out was to see the analogy between an "element of a subset" and a "distinction of a partition," where a _distinction_ or _dit_ of a partition is an ordered pair of elements of the underlying set \\(X\\) that are in different blocks or parts of the partition. The _indistinctions_ or _indits_ of a partition are the ordered pairs of elements in the same block or part of the partition. Hence the set of all indits of a partition is just the binary equivalence relation associated with the partition, and the set of all dits of a partition, the _ditset_ \\(dit(P)\\), is the complementary binary relation called an _apartness relation_ or just a _partition relation_. The _logical entropy_ \\(h(P)\\) of a partition is just the normalized size of the ditset:

$$h(P)=\frac{|dit(P)|}{|X^2|}.$$

Logical entropy is related in a definite way to Shannon entropy. All the Shannon definitions of simple, joint, conditional, and mutual entropy can be obtained by a uniform requantifying transformation from the corresponding definitions in logical information theory. Moreover, logical entropy (unlike Shannon entropy) is a measure in the sense of measure theory, and thus the logical definitions of simple, joint, conditional, and mutual logical entropy of partitions is just the measure applied respectively to the ditsets, the union of ditsets, the difference of ditsets, and the intersection of ditsets. Indeed, logical entropy is a probability measure with the interpretation that \\(h(P)\\) is the probability of getting a distinction in two independent draws from the set \\(X\\). The basic paper developing logical information theory (and spelling out the comparisons with the closely related Shannon theory) was also published in the [Logic Journal of the IGPL](http://www.ellerman.org/logical-information-theory/).

I went into a little detail about logical information theory as the quantitative version of partition logic since it bears on the question of which of the opposite partial orderings, the coarseness ordering or the refinement ordering, should be used on partitions. Since refinement increases the number of distinctions, it increases logical (and Shannon) entropy so the refinement ordering seems best if one is emphasizing the ordering of partitions according to their information content measured by logical or Shannon entropy.

Another topic of my research has been "heteromorphisms" which are morphisms between the objects of different categories, e.g., the canonical injection of the set of generators into the free group on that set. Homomorphisms may be treated formally by hom-functors, and heteromorphisms are treated formally by the "het-functors" or _profunctors_ that appear later in the course. For one reason or another, Mac Lane and Eilenberg did not include heteromorphisms in their basic "ontology" for category theory even though they are just as much a tool for the proverbial "working mathematician" (referenced in the title of Mac Lane's text) as are homomorphisms. The French school of category theory, e.g., the Grothendieck school, is much less prudish and let hets 'out of the closet' as just "morphisms" routinely. The differing attitude about hets comes out in the treatment of adjoint functors. The simpler and I think more natural treatment of adjoints was developed by Bodo Pareigis in the late 60's and published in his text translated into English as: Pareigis, Bodo. 1970. _Categories and Functors_. New York: Academic Press. It also comes out in what one takes as more important or more basic: adjoint functors or representable functors. Grothendieck gives representable functors the pride of place, whereas the American school puts the emphasis on adjoint functors. Hets, representable functors, and adjoints are all tied together in the Pareigis treatment of adjoints. Take the het-functor whose value \\(\text{het}(X,G)\\) is the set of set-to-group maps from a set \\(X\\) to a group \\(G\\). The underlying set functor is just the right-representation of those hets by homs in the category of sets, and the free-group functor is just the left-representation of those _same hets_ by homs in the category of groups. Thus one has the natural isomorphisms:

$$ \text{hom}(F(X),G)\cong \text{het}(X,G)\cong \text{hom}(X,U(G)).$$

The heterophobic American school always leaves out the middle term of hets and just defines an adjunction using the natural isomorphism between the hom-sets. All left- and right-representations of a het-functor define a pair of adjoint functors, and given a pair of adjoint functors, there is a het-functor so that the given adjoints are isomorphic to the left- and right-representations of the het-functor. All this holds as well for adjunctions on preorders or Galois connections. An introductory paper is [here](http://www.ellerman.org/adjoint-functors-and-heteromorphisms/).

The quantitative version of Boole's logic of subsets is just finite logical probability theory as developed by Boole in his _Laws of Thought_. Rota always held that probability (i.e., the normalized size of a subset) is to subsets as information is to partitions.

$$\frac{\text{Probability}}{\text{Subsets}}=\frac{\text{Information}}{\text{Partitions}}$$

The key to working that out was to see the analogy between an "element of a subset" and a "distinction of a partition," where a _distinction_ or _dit_ of a partition is an ordered pair of elements of the underlying set \\(X\\) that are in different blocks or parts of the partition. The _indistinctions_ or _indits_ of a partition are the ordered pairs of elements in the same block or part of the partition. Hence the set of all indits of a partition is just the binary equivalence relation associated with the partition, and the set of all dits of a partition, the _ditset_ \\(dit(P)\\), is the complementary binary relation called an _apartness relation_ or just a _partition relation_. The _logical entropy_ \\(h(P)\\) of a partition is just the normalized size of the ditset:

$$h(P)=\frac{|dit(P)|}{|X^2|}.$$

Logical entropy is related in a definite way to Shannon entropy. All the Shannon definitions of simple, joint, conditional, and mutual entropy can be obtained by a uniform requantifying transformation from the corresponding definitions in logical information theory. Moreover, logical entropy (unlike Shannon entropy) is a measure in the sense of measure theory, and thus the logical definitions of simple, joint, conditional, and mutual logical entropy of partitions is just the measure applied respectively to the ditsets, the union of ditsets, the difference of ditsets, and the intersection of ditsets. Indeed, logical entropy is a probability measure with the interpretation that \\(h(P)\\) is the probability of getting a distinction in two independent draws from the set \\(X\\). The basic paper developing logical information theory (and spelling out the comparisons with the closely related Shannon theory) was also published in the [Logic Journal of the IGPL](http://www.ellerman.org/logical-information-theory/).

I went into a little detail about logical information theory as the quantitative version of partition logic since it bears on the question of which of the opposite partial orderings, the coarseness ordering or the refinement ordering, should be used on partitions. Since refinement increases the number of distinctions, it increases logical (and Shannon) entropy so the refinement ordering seems best if one is emphasizing the ordering of partitions according to their information content measured by logical or Shannon entropy.

Another topic of my research has been "heteromorphisms" which are morphisms between the objects of different categories, e.g., the canonical injection of the set of generators into the free group on that set. Homomorphisms may be treated formally by hom-functors, and heteromorphisms are treated formally by the "het-functors" or _profunctors_ that appear later in the course. For one reason or another, Mac Lane and Eilenberg did not include heteromorphisms in their basic "ontology" for category theory even though they are just as much a tool for the proverbial "working mathematician" (referenced in the title of Mac Lane's text) as are homomorphisms. The French school of category theory, e.g., the Grothendieck school, is much less prudish and let hets 'out of the closet' as just "morphisms" routinely. The differing attitude about hets comes out in the treatment of adjoint functors. The simpler and I think more natural treatment of adjoints was developed by Bodo Pareigis in the late 60's and published in his text translated into English as: Pareigis, Bodo. 1970. _Categories and Functors_. New York: Academic Press. It also comes out in what one takes as more important or more basic: adjoint functors or representable functors. Grothendieck gives representable functors the pride of place, whereas the American school puts the emphasis on adjoint functors. Hets, representable functors, and adjoints are all tied together in the Pareigis treatment of adjoints. Take the het-functor whose value \\(\text{het}(X,G)\\) is the set of set-to-group maps from a set \\(X\\) to a group \\(G\\). The underlying set functor is just the right-representation of those hets by homs in the category of sets, and the free-group functor is just the left-representation of those _same hets_ by homs in the category of groups. Thus one has the natural isomorphisms:

$$ \text{hom}(F(X),G)\cong \text{het}(X,G)\cong \text{hom}(X,U(G)).$$

The heterophobic American school always leaves out the middle term of hets and just defines an adjunction using the natural isomorphism between the hom-sets. All left- and right-representations of a het-functor define a pair of adjoint functors, and given a pair of adjoint functors, there is a het-functor so that the given adjoints are isomorphic to the left- and right-representations of the het-functor. All this holds as well for adjunctions on preorders or Galois connections. An introductory paper is [here](http://www.ellerman.org/adjoint-functors-and-heteromorphisms/).