# In category theory, why do we meet more left adjoints than right adjoints

In this answer, the author states that "many of the naturally occurring functors we meet tend to have left adjoint but often they lack right adjoints".

Is there any philosophical explanation to this fact?

• @JoWehler -- there is an arbitrary convention. That convention does not have a theoretical basis, but it has an actual asymmetry within observed math. Hence the question! Commented Feb 28, 2022 at 21:40
• @JoWehler Sure! But in practice, the right adjoints often come first.
– Bob
Commented Feb 28, 2022 at 21:42
• Per WP: The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F... So in most application of math we're more interested in finding a most efficient solution (left adjoint) to some problem, not problem itself, so philosophically explains why one meets left ajdoints more often... Commented Mar 1, 2022 at 7:00
• @DoubleKnot This might be the beginning of a nice answer. At the moment, the proposed answers are not talking about adjunction in particular, but about symmetry in general.
– Bob
Commented Mar 1, 2022 at 7:33
• @Dcleve But if logic is contingent, then the sentence "logic is contingent" is neither true nor false! Or do you assume a meta-logic that is necessary?
– Bob
Commented Mar 3, 2022 at 7:09

Short answer: We tend to study inclusions.

Many inclusions are naturally-occurring right adjoints. Consider an arbitrary subcategory relationship; the functor including the subcategory into the supercategory is right adjoint, and its left adjoint is then some sort of free functor constructed by the mentioned adjoint functor theorems.

First, why aren't inclusions left adjoints? Well, they can be, but it's unlikely. All colimits must be preserved, including sums/coproducts. Speaking in a very rough and informal manner (so, philosophically speaking!) we could imagine that we have a direct sum of two objects in the subcategory, and we want to map it to a direct sum of two objects in the supercategory. But we are generalizing these objects when we move to a bigger category with looser axioms, and two specialized objects may no longer be clearly distinct after generalization, so the direct sum may not be preserved. This principle frustrates construction of right adjoints to inclusion functors.

Second, why are inclusions so common/important/studied? Inclusions give a category-theoretic "formally formal" veneer to the is-a principle. For example, the inclusion AbGrp can be read, "every Abelian group is a group" and captures quite a bit of what we might (philosophically!) mean by that, including the idea that transformations of Abelian groups are transformations of groups, etc. More dramatically, there is no 2-inclusion from DagCatCat or vice versa; dagger-categories are not categories, nor vice versa!

Finally, why doesn't duality apply to all of this? Because the subcategory relation in any 2-category ought to -- in the sense of Cheng morality -- yield a poset. Posets aren't self-dual.

In this answer, the author states that "many of the naturally occurring functors we meet tend to have left adjoint but often they lack right adjoints".

Is there any philosophical explanation to this fact?

Philosophy can provide a partial explanation and conjecture, but scientific explanation would require psychology, linguistics, and mathematical thinking.

### Cognition and Norms

From the philosophy of psychology, we have a basic fact of sorts, and that is that cognition has bias. In fact, cognition is inherently normative. This is a shock to many thinkers in the Western world because the sciences have strived so hard to be objective. But, even after the logical positivists attacked normativity with full force, the outcome was that all theory is laden with normativity. So, the first thing philosophy offers as an insight is that according to philosophy of mind and philosophy of psychology, biases, values, and norms ARE the norm.

The fact that mathematical discourse might favor the expression of one syntactical expression over another is a common example of bias. The exact nature of any cognitive bias is generally studied by the science built on the respective philosophy. For instance, in the philosophy of language, one studies the nature of the syntax of writing systems, but it is linguists who more generally study directionality in writing systems. In this case, psychologists and linguists would be those who could provide empirical evidence to support an argument as to why one adjoint is seen in the literature more than another. Philosophy helps provide metaphysical language to undergird the sciences, which is why for every science, there is a philosophy of science. This might be even seen as support for Kuhn's ideas about paradigmatic and normal science. But let's look at the specific question related to the antisymmetry of the graphemes in question.

### Biases in Irreflexive Syntax

In the math world, what we are talking about more generally is the reflexive and irreflexive. A simpler example might be more enlightening. Consider the use of the material conditional which draws from the same family of allographs, right and left arrows of various sorts, as the adjunction of categories. In logic, one can write:

A -> B

And this is read as 'A implies B' or 'if A then B' and so on. Pick up a logic textbook and the textbook will be stuffed with formal logic that uses the character right-arrow for arguments. Modus ponens and modus tollens are two easy examples. And yet, there's no reason, other than convention and habit that the conditional must be written from left to right. One can write:

B <- A

In fact, we use the natural language 'B because A' all of the time. 'I went home because it was raining.', 'I got fired because I didn't go to work.', and 'I have a hard time waking up because I go to sleep late.' are quite natural. And yet, I can't recall seeing modus ponens as:

Q <- P
P
Therefore Q

Why not? Why is there a bias in writing the conditional with the right arrow instead of the left arrow?

### Biases in Linguistic Syntax

Well, philosophy can speculate, but philosophical conjectures become scientific when they become very rational and empirical. In the case, linguistics may shed some light. It is a well known fact that there are biases in word order in languages. In fact, while older languages often use declension and conjugation to supplement the semantics of a language, as languages age and blend, they frequently rely on word order to convey meaning. In Latin, 'Puer ama ranas' and 'Ranas ama puer' both mean 'The boy loves frogs.' In English, if one reverses the subject and object, then the lover and that-loved changes. Here again, a bias in direction!

So, to review, we have biases in graphemes, the directionality or our texts, and the word order of our sentences, and philosophy tackles the commonality to this bias, and helps to flesh in the presumptions of the various fields such as mathematics, psychology, and linguistics. But on these sort of empirical questions, it largely remains silent. So who then?

### How Philosophies Support the Sciences

Well, if logicians use right-arrows more than left ones or category theorists use left adjoints more than right ones, that seems par for the course. But as to the the why, that's a question for psychologists and linguists who study the behavior of mathematicians, because philosophy over the last few hundred years went from natural philosophy to the sciences. And the sciences might be thought of as providing pre-chewed philosophical thinking, so that instead of a scientist having to master ontology, epistemology, and axiology, a scientist can learn some math, some empirical heuristics, and get busy to building experiments for establishing high degrees of correlation and intervals of confidence. And explanations that rely on peer-review, rigorous logic, empirical evidence, and mathematical techniques are certain in a way that haphazard metaphysical speculation are not. So strictly speaking, philosophy could provide an answer to your question, but it is philosophy rooted in the philosophy of science which is likely to be a good answer. (And yes, I am apologetically scientistic on this account. Not all ideas are good for survival.)

As to the specific question, why left adjoint functors? I would suspect the same reason the material conditional points right: English and the popular PIE languages are SVO languages largely, and the left adjoint functor merely reflects that bias from left to right. If one writes the verb with the right-arrow, SVO is simply 'S->O'. Of course, to arrive at a scientific answer, a study would have to be devised to see if in other cultures that are not SVO, a different sort of bias could be found.

• This is interesting, but I can't see how it relates to my question.
– Bob
Commented Mar 1, 2022 at 20:13
• You said that left adjoints occur more than right ones in the literature. I'm simply asserting that it might be something as a syntactical bias and explaining to you why you won't get a definitive answer in Philosophy SE. :D If you're digging for some grand speculative metaphysical conjecture that it has to do with a fundamental property of the universe, you might be overthinking why one mathematical object is favored over it's partner in symmetry. Did I misunderstand your question?
– J D
Commented Mar 1, 2022 at 20:30
• Philosophy is rife with people who think that grammatical phenomena are some sort of magical properties of the universe. Words are just sounds. Writing is just ink.
– J D
Commented Mar 1, 2022 at 20:31

Since there an infinite number of possible categories, morphisms, etc., the reality is that there aren't more left adjoints than right adjoints all things considered (hence the answer you link to, on the MathSE, says that it's "nonsense" to say that there are more); it's just that, per the qualities of most relevant functors that have actually been written about in detail, the discrepancy occurs (hence what the linked-to answer says about category theory vs. "practiced mathematics").

But so the reason for the discrepancy is localized more to the functors under discussion; there's no universal law like, "More functors have left adjoints than right adjoints for some all-encompassing reason." It's not like there are ℵ10-many functors in general such that only ℵ1-many have right adjoints and the rest have only left adjoints. Let's say (not technically correctly at all) that there are, however, only ℵ10-many functors that have been studied in great detail to date, and only ℵ1-many of those, called "naturally occurring," happen to have right adjoints.

That description "naturally occurring" shoulders most of the dialectical weight, here: offhand, I'd assume they mean "occurring in contexts like lower-level algebra, geometry, topology, etc. instead of in the context of category theory generally." The other answers to this question here on the PhilosophySE then get at why the "naturally occurring" functors exhibit the relevant asymmetry: because we identify those functors in connection with less abstruse, and more potentially empirically situated, questions (and symmetry/asymmetry is all the rage in e.g. physics).

Emily Riehl in Category Theory in Context appears to provide an inverse half of an explanation: on pg. 116 she writes:

The basic calculus of adjunctions is developed in §4.4, with each result proven twice, once using the Yoneda lemma and again via the techniques of “formal category theory,” to illustrate two important methods of categorical reasoning. In §4.5, we prove what is perhaps the most frequently applied result in category theory: that right adjoint functors preserve limits while left adjoint functors preserve colimits. This explains why limits tend to be easier to construct than the formally dual colimit notions.

I have very little personal understanding of what I'm reading, here, so it sounds to me as if we would meet with right adjoints more often on account of the easier-than condition Riehl mentions, so I know I'm not interpreting this passage correctly on my own terms, but maybe you will interpret it correctly and in this understand how it at least should help to answer your question.

"Mathematically' they say that this assertion is 'nonsense'. And pactically, they say its because mathematics in practise in not 'self-dual'.

This is akin to the observation that although formal mathematics allows any consistent set of axioms as a description of a formal system; in practise, the systems that mathematicians use grow organically out of their experience. Thus, they say these formal systems are 'natural' or 'found in nature'. Whilst an artificial set of axioms is just that - artificial and not natural and probably 'ugly' to boot.

We see the same phenomena in physics, Newtonian mechanics is manifestly symmetric in time; nevertheless, time is real, moves 'forward' and is open. Practical physics is asymmetric with time.

Duality is a symmetry, and whilst mathematicians like symmetry, they are well aware that mathematics, like physics, like nature is not wholly symmetric and can be asymmetric. To use a physic's term, symmetry is broken in nature and also in natural mathematics.

• Are you claiming that mathematics is asymmetric because the world is asymmetric and mathematicians took their inspiration from the world?
– Bob
Commented Feb 28, 2022 at 21:20
• @Bob: As long as your 'world' includes the world of mathematical experience .. Commented Mar 1, 2022 at 1:34

Yes, philosophy has an answer. Our universe is contingent. There are many possible ways it could have operated, in principle. What math, or logic applies to it in practice, cannot be predicted from logic principles, or math theory.

This point was articulated with great force by Kant in The Critique of Pure Reason. Kant however, did not go as far as philosophy has today. Kant held there were at least a few math relations we could know a priori that were true, and his go-to example was Euclidean geometry. Subsequently, non-Euclidean geometries were discovered in potential math space, and they actually turn out to be how our universe behaves! The desire to believe in the a priori "truth" of Euclidean geometry turned out to be a fallacy, of failure of imagination on Kant's part (the informal fallacy is generally called argument from ignorance)!! Apply this lesson to our world, and we cannot predict a priori (from logic or math principles) what math will apply to any physical aspect of our universe .

Subsequently, Godel discovered that one cannot even derive logic or math principles with confidence even within math-space, much less whether they apply to our world. And logic pluralism shows that there are infinite logics, and we cannot know if any of them even apply to our world. https://math.vanderbilt.edu/schectex/logics/

So -- the answer philosophy has, is that the lack of symmetry that mathematicians have found in applying math to our world, just happens to be the way our contingent world worked out.

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