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# About "category" in everyday language

"Category" is a pretty common word in everyday language. For example:

1. On the left side of this forum we see the grouping "Categories" which subsume various hyperlinked items such as "All Categories", "Applied Category Theory Course", "Applied Category Theory Exercise", etc.
2. In English dictionaries, we see examples like "the various categories of research" (Oxford), "taxpayers fall into one of several categories" (Merriam-Webster), "I wouldn't put this book in the same category as the author's first novel" (Wiktionary), etc.
3. In other disciplines (less everyday), such as Aristotle's categories in philosophy, grammatical categories in linguistics, etc.

I wondered if such non-mathematical uses of "category" could be given some category-theoretic modeling. Considering we already have a category Cat for all categories, might there also a category for all uses of the word "category"? Or is it simply part of Cat?

I'm guessing such a modeling, if possible, wouldn't involve too much technical complication - perhaps just sets - but it'd be interesting to have a CT perspective to the issue. :-)

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1.
edited May 2018

I wondered if such non-mathematical uses of "category" could be given some category-theoretic modeling. Considering we already have a category Cat for all categories, might there also a category for all uses of the word "category"? Or is it simply part of Cat?

Sadly, I don't think so. Category is just a term that mathematicians made up. It's an overloaded term, however. Aristotle used it one way as you mention. Categorical distributions in statistics refer to something else. Kant's categorical imperative is something yet different still.

Keeping the terminology straight is hard. When a computer scientist talks about a network topology, she doesn't mean general topology. She usually means a graph structure. When a cryptographer talks about lattice cryptography, they don't care about lattices from lattice theory. And λ-calculus isn't the calculus we learned in high school.

Comment Source:> I wondered if such non-mathematical uses of "category" could be given some category-theoretic modeling. Considering we already have a category Cat for all categories, might there also a category for all uses of the word "category"? Or is it simply part of Cat? Sadly, I don't think so. *Category* is just a term that mathematicians made up. It's an overloaded term, however. Aristotle used it one way as you mention. [Categorical distributions](https://en.wikipedia.org/wiki/Categorical_distribution) in statistics refer to something else. [Kant's categorical imperative](https://en.wikipedia.org/wiki/Categorical_imperative) is something yet different still. A similar confusing word thrown around is *monad*. It started as an idea in ancient Greek philosophy. Parmenides thought of his *monad* as the totality of all things. In Leibniz's [*Monadology* (1714)](https://en.wikipedia.org/wiki/Monadology) he was referring to sort of elementary, indivisible substance. Centuries later, the mathematician Abraham Robinson took inspiration from Leibniz's approach to calculus in devising [non-standard analysis](https://en.wikipedia.org/wiki/Non-standard_analysis). Robinson called one mathematical construct he invented a [*monad*][monad], but it has nothing to do with Leibniz's monads as far as I can tell. The term is so confusing that Goldblatt just calls them *halos* in his 1998 text *Lectures on the Hyperreals*. Saunders Maclane coined *monad* as it is commonly used in category theory about a decade later. It has nothing to do with Robinson's monad. In the late 80s Philip Wadler introduced monads to Haskell to model side effects and it's been baffling programmers ever since. And yet still there's [*monadic first order logic*](https://en.wikipedia.org/wiki/Monadic_predicate_calculus) where *monadic* is just Greek for every predicate can take at most one argument. Keeping the terminology straight is hard. When a computer scientist talks about a network topology, she doesn't mean [general topology](https://en.wikipedia.org/wiki/General_topology). She *usually* means a graph structure. When a cryptographer talks about [lattice cryptography](https://en.wikipedia.org/wiki/Lattice-based_cryptography), they don't care about lattices from [lattice theory][lattice theory]. And λ-calculus isn't the calculus we learned in high school. [monad]: https://en.wikipedia.org/wiki/Monad_(non-standard_analysis) [lattice theory]: https://en.wikipedia.org/wiki/Lattice_(order)
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edited May 2018

Thanks for the very informative comment, @Matthew! Indeed, very often disciplinary concepts could have been given less overloaded terminology (e.g. I can totally feel your "calculus" point!). Apart from this perhaps socio-historical matter, though, what about the non-technical usage of the word "category" (in the sense of a class/type I guess)? Since many people don't have training in CT/philosophy/logic, the terminological overloading is irrelevant for them, but the "category" in their vocabulary - due to its sense - may still correspond to something mathematically modelable, e.g. as a set or an equivalence class, and since when a certain "category" is mentioned it is usually presupposed that several distinct "categories" exist on the given matter (e.g. a book may belong to one "category" or another), perhaps what's at issue here is the organization of "categories" per se (hope I'm not talking nonsense...)?

On that note, I realize my initial question about the entire class of different uses of a word (of which "category" is an example) could easily drive me to the direction of lexical semantics, so perhaps it wasn't a meaningful pursuit in our CT context anyway... (or is it?) :P

Comment Source:Thanks for the very informative comment, @Matthew! Indeed, very often disciplinary concepts could have been given less overloaded terminology (e.g. I can totally **feel** your "calculus" point!). Apart from this perhaps socio-historical matter, though, what about the non-technical usage of the word "category" (in the sense of a class/type I guess)? Since many people don't have training in CT/philosophy/logic, the terminological overloading is irrelevant for them, but the "category" in their vocabulary - due to its sense - may still correspond to something mathematically modelable, e.g. as a set or an equivalence class, and since when a certain "category" is mentioned it is usually presupposed that several distinct "categories" exist on the given matter (e.g. a book may belong to one "category" or another), perhaps what's at issue here is the organization of "categories" per se (hope I'm not talking nonsense...)? On that note, I realize my initial question about the entire class of different uses of a word (of which "category" is an example) could easily drive me to the direction of [lexical semantics](https://en.wikipedia.org/wiki/Lexical_semantics), so perhaps it wasn't a meaningful pursuit in our CT context anyway... (or is it?) :P 
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3.

I believe Lawvere tried to connect the meanings across philosophy and math. The same with "natural transformation" and "functor".

In fact, the words "category", "functor", and "natural transformation" were swooped from philosophy if memory serves me.

Comment Source:I believe Lawvere tried to connect the meanings across philosophy and math. The same with "natural transformation" and "functor". In fact, the words "category", "functor", and "natural transformation" were swooped from philosophy if memory serves me.
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@Keith Yeah I noticed the terminological overlapping between CT and philosophy/linguistics too - I had initially thought we borrowed from CT but it turned out to be the other way around. :P Do you perhaps know where Lawvere made the philosophy–math connection? I would be interested to read on that. :)

Comment Source:@Keith Yeah I noticed the terminological overlapping between CT and philosophy/linguistics too - I had initially thought we borrowed from CT but it turned out to be the other way around. :P Do you perhaps know where Lawvere made the philosophy–math connection? I would be interested to read on that. :)
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5.

I don't recall off hand the exact text that came from.

Comment Source:I don't recall off hand the exact text that came from.
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edited June 2018

I agree with Matthew that pushing the parallelism is a bad idea. In Aristotle I recall that categories were supreme genera, maximal elements in an order with resemblance to proper classes in set theory but not much fruitful analogy with modern ones comes to mind. In the case of syntactic categories in liguistics the situation is worse because it seeds confusion: some times one needs to speak simultaneously of syntactic categories in the context of a mathematically categorical treatment as did Lambek and there are interferences when merging two traditions (linguists and mathematicians). I think now linguists say categorial and mathmen categorical to mark distinctions, as in combinatory categorial grammar.

Comment Source:I agree with Matthew that pushing the parallelism is a bad idea. In Aristotle I recall that categories were supreme genera, maximal elements in an order with resemblance to proper classes in set theory but not much fruitful analogy with modern ones comes to mind. In the case of syntactic categories in liguistics the situation is worse because it seeds confusion: some times one needs to speak simultaneously of syntactic categories in the context of a mathematically categorical treatment as did Lambek and there are interferences when merging two traditions (linguists and mathematicians). I think now linguists say categorial and mathmen categori**cal** to mark distinctions, as in combinatory categorial grammar.
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7.

Matthew, I'm an imperative languages guy (did some lisp ages ago...) but find itchy that in your enumeration you seem to put Haskellite monads in other shelf than the category-theory ones. I had the vague impression that in Haskell there was more or less the intention to reflect the mathematical idea (as in say Moggi), even if in some dialectal form or obeying language servitudes.

Comment Source:Matthew, I'm an imperative languages guy (did some lisp ages ago...) but find itchy that in your enumeration you seem to put Haskellite monads in other shelf than the category-theory ones. I had the vague impression that in Haskell there was more or less the intention to reflect the mathematical idea (as in say Moggi), even if in some dialectal form or obeying language servitudes.
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8.

How about this one concerning the definition of functor -- in Prolog, everything is defined as a structure or term of the form: functor(arg1, arg2, ..).

Functors are thus the Prolog building blocks to declaratively relate or map different categories.

Comment Source:How about this one concerning the definition of functor -- in Prolog, everything is defined as a structure or term of the form: functor(arg1, arg2, ..). Functors are thus the Prolog building blocks to declaratively relate or map different categories.
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9.

Aw yes, now I remember. It was on nlab's category theory page.

https://ncatlab.org/nlab/show/category+theory

Specifically:

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.

(Saunders Mac Lane, Categories for the Working Mathematician, 29–30).

Comment Source:Aw yes, now I remember. It was on nlab's category theory page. https://ncatlab.org/nlab/show/category+theory Specifically: >Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance. >(Saunders Mac Lane, Categories for the Working Mathematician, 29–30).
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10.
edited October 2018

A point of history, Wadler did not introduce monads in PL's, Moggi and Plotkin did.

Comment Source:A point of history, Wadler did not introduce monads in PL's, Moggi and Plotkin did.
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11.

@Keith Yeah I came across that paragraph too when reading Mac Lane's textbook. :-)

It's been so long since I last logged in (due to overwhelming workload...) but now that I have a bit more time I'm determined to catch up with everything that happened here while I went on estivation! :P

Comment Source:@Keith Yeah I came across that paragraph too when reading Mac Lane's textbook. :-) It's been so long since I last logged in (due to overwhelming workload...) but now that I have a bit more time I'm determined to catch up with everything that happened here while I went on estivation! :P
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Matthew, I'm an imperative languages guy (did some lisp ages ago...) but find itchy that in your enumeration you seem to put Haskellite monads in other shelf than the category-theory ones. I had the vague impression that in Haskell there was more or less the intention to reflect the mathematical idea (as in say Moggi), even if in some dialectal form or obeying language servitudes.

Comment Source:> Matthew, I'm an imperative languages guy (did some lisp ages ago...) but find itchy that in your enumeration you seem to put Haskellite monads in other shelf than the category-theory ones. I had the vague impression that in Haskell there was more or less the intention to reflect the mathematical idea (as in say Moggi), even if in some dialectal form or obeying language servitudes. Haskell monads are just monads on the category Hask (which is a haskell programing term for the somewhat nebulous category of haskell types and functions).