Act only according to that maxim by which you can at the same time will that it should become a universal law.
Immanuel Kant (1785)
Precursors Of Category Theory
Peirce
Cued by Kant’s idea on the function of concepts in general, Peirce locates his categories on the highest level of abstraction affording a meaningful measure of traction in practice, reserving judgment on the absolute unity of perfect ambiguity and the numerous dualisms which taken together may well converge on the same conception as Peirce’s trinity.
Selection 1
§1. This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it. (CP 1.545).
§2. This theory gives rise to a conception of gradation among those conceptions which are universal. For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied; and so on. (CP 1.546).
C.S. Peirce, “On a New List of Categories” (1867)
Selection 2
I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.
That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of. We thus think of the thought-sign itself, making it the object of another thought-sign.
Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions. Does this series proceed endlessly? I think not. What then are the characters of its different members?
My thoughts on this subject are not yet harvested. I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being: Actuality, Possibility, Destiny (or Freedom from Destiny).
On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being. Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments. (CP 4.549).
C.S. Peirce, “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546 (1906), CP 4.530–572.
The first thing to extract from this passage is that Peirce’s Categories, or “Predicaments”, are predicates of predicates. Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the logic of second intentions, or what is handled very roughly by second order logic in contemporary parlance, and continuing onward through higher intentions, or higher order logic and type theory.
Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel. The names he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like quality, reaction, and symbolization to maximally abstract terms like firstness, secondness, and thirdness. Taken in full generality, k-ness may be understood as referring to those properties all k-adic relations have in common. Peirce’s distinctive claim is that a type hierarchy of three levels is generative of all we need in logic.
Part of the justification for Peirce’s claim that three categories are necessary and sufficient appears to arise from mathematical facts about the reducibility of k-adic relations. With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates. With regard to sufficiency, all higher arity k-adic relations can be analyzed in terms of triadic and lower arity relations.
Inquiry Into Inquiry 
My reply to a reader’s question on Facebook may be worth including here —
“I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.” (CP 4.549).
That is a difficult passage, partly because Peirce conducts the discussion in dialogue form, so one may question whether he is putting a variant of his own words in the mouth of the other or merely using the other as a foil to be corrected. I have chosen to go with the first option until some evidence proves it untenable.
The word monadic is a predicate of predicates. It holds of all predicates that express a simple quality of things. The comprehension or intension of a monadic predicate is a property that things can have independently of reference to other things. The extension of a monadic predicate is the set of things that have that property in common.
Likewise, dyadic is a predicate of predicates, holding of all 2-place predicates, properties, or relations, and triadic is a predicate of predicates, holding of all 3-place predicates, properties, or relations.
In their purely mathematical aspect, Firstness, Secondness, and Thirdness are simply what all monadic, dyadic, and triadic relations, respectively, have in common. Of course the plot thickens when we turn to phenomenology because that requires us to ask whether there is anything about the phenomenon in view that makes a particular relational model more apt than any other.
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