Johnson He 2024 5 5 2025 9 27 https://hejohns.github.io/forest/index/ index /forest/index/ Johnson He false Johnson He20246102025927https://hejohns.github.io/forest/hejohns-000O/hejohns-000O/forest/hejohns-000O/About Me I am a PhD student in Computer Science at Indiana University. My advisor is Carlo Angiuli. I'm interested in logic and related areas of computer science, particularly programming languages. This is my forest. email: photo: For more information about me, see Johnson He. 2024552025927Notes202411112025927https://hejohns.github.io/forest/hejohns-000V/hejohns-000V/forest/hejohns-000V/ACL permissionsExposition I thought I needed (extended) ACL permissions earlier (in the end, it was a different problem), and needed to brush up on how they work. POSIX Access Control Lists on Linux has a nice description of ACL and how it interacts with traditional owner-group-other 3-3-3 permissions. I was surprised that named user ACL entries "are assigned to the group class" (as well as named group entries and the owning group entry). Naïvely, I would've assumed they'd be in the user class, but it looks like the user class is just the owning user entry. I'm not yet convinced of a fundamental design flaw with the naïve semantics-- and having a user mask-- although of course it would violate the Unix ownership model of having a unique owning user and group. Should people expect that chmod 700 means you're the only user than can access the file? Maybe the use-case seems silly, but I wanted two users to share a directory and all contained files and subdirectories (hereditarily, like setgid). Ie, I wanted truly group ownership. This has to have a canonical solution? It seems like the obvious closest solution is to create a group for the two users and setgid the directory so everything hereditarily is owned by the group. But the "one-true-owner" (not a quote) semantics means that you fundamentally can't get this to work, and allows the owner to chmod g- (reduce the group permissions relative to the "true" owner), leaving the non-true-owner with lower permissions. No fiddling with default ACLs can prevent this. For truly group ownership, both users should have the same view. More specifically, they should have the same access, or else one user can obliviously proceed happily while the other is denied permission. In my case earlier, ExtUtils::MakeMaker chmod 755'd the build directory, so as the "true" owner, I could rwx, but the other user could only r-x, causing make and make clean to error part-way, leaving them stuck. Johnson He20248202025927https://hejohns.github.io/forest/hejohns-000T/hejohns-000T/forest/hejohns-000T/Brushing up on Acronyms, Reprise: GATs, SOGATs, or even just Algebraic TheoriesExpositionJohnson He20248202025927intro I don't know where to start with all this. Over the last couple months, I keep hearing about Taichi Uemura and SOGATs, or CwRs (categories with representable maps) and how they're a modern natural model or something or other. And something something initiality conjecture, and I'm left wondering what this all is such that the existence of a free structure is so nontrivial? If you know me, you know my style is to paste together newspaper clippings of references and use this tree as a sort of digital collage. So let's begin with what's probably the most up-to-date perspective: Johnson He20248202025927Uemura's Thesis §4 (SOGATs) "In this chapter, we introduce second-order generalized algebraic theories (SOGATs) for several reasons. In short, SOGATs are: 1. syntactic counterparts of CwRs; 2. extension of the general definition of dependent type theories given by Bauer, Haselwarter, and Lumsdaine [20] to allow user-definable judgment forms; 3. second-order extension of generalized algebraic theories of Cartmell [35]; 4. dependently-typed extension of second-order algebraic theories of Fiore and Mahmoud [57]. The first aspect is the most important in this thesis. In [169] the author introduced a logical framework to give syntactic counterparts of CwRs, but it is not satisfactory. When defining a type theory as a theory over a logical framework, one has to prove the adequacy of the definition, that is, the theory over the logical framework derives the same judgments as the original type theory does. Although adequacy can systematically be proved for each individual type theory syntactically [75] or semantically [79], the adequacy of all logical framework presentations cannot be proved nor even stated unless we have a general notion of a type theory outside the logical framework. What we really want is a general notion of a syntactic presentation of a type theory such that a wide range of existing type theories are verified to be instances of that notion by just unfolding the definition. Bauer, Haselwarter, and Lumsdaine [20] have made a careful analysis on inference rules in traditional presentations of type theories and proposed a class of syntactic presentations of type theories that fits our requirement. A restriction on their type theories is that the forms of judgments are fixed, and thus we cannot define some complex type theories such as cubical type theory [41]. We thus modify their definition of type theories to allow users to declare new judgment forms. It turns out that our syntactic presentations of type theories are considered as an extension of algebraic theories by dependent types and variable binding. We thus choose the name 'second-order generalized algebraic theory' combining the dependently-typed extension and the second-order extension of algebraic theories found in the literature." Maybe for the cognoscenti this makes perfect sense, but for me: re 1. What does "syntactic counterpart" mean? I suppose now's a great time to go back to §3 and actually learn what a CwR is in the first place, but if it's anything like a natural model, isn't it already very syntactic? Maybe this is totally off, but natural models/CwFs look a lot like hyperdoctrines in the sense that they directly model the syntax. What does it mean to have a syntactic counterpart to something like a hyperdoctrine, that's already the syntactic model? Or is it roughly like (regular) hyperdoctrine : (regular) category :: SOGAT : CwR? Uemura's further explanation of 1. seems to suggest I have no idea what "syntactic" means here. re 2. I'm also going to have to follow this reference, but "A restriction on their type theories is that the forms of judgments are fixed, and thus we cannot define some complex type theories such as cubical type theory" indeed seems like a major shortcoming. re 3. Maybe the original syntactic-thing-for-dependent-type-theories. It seems necessary to understand this fundamental notion at some point. re 4. And this of course suggest brushing up on what an algebraic theory even is in the first place-- what's the relationship with Lawvere theories? Johnson He20248202025927Uemura's Thesis §3 (CwRs) "In this chapter, we introduce the notion of a category with representable maps (CwR) as an abstract notion of a type theory. This notion covers a wide range of syntactically presented type theories as we will see in Chapter 4 and has nice semantic properties proved in Chapter 5. The idea comes from Lawvere’s functorial semantics of algebraic theories [108] and its variants [123, 1]. In functorial semantics, theories are identified with struc- tured categories, and models of a theory are identified with structure-preserving functors from the structured category corresponding to the theory. For example, consider the theory of groups whose models are of course groups. A group is a set A equipped with three operators satisfying certain equational axioms. The operators form the following diagram of sets. [I've changed the 1 to e because of rendering issues I can't figure out.] In category theory, a diagram is nothing but a functor, and thus a group would be a functor from a suitable category . What structure should have? The category should contain a diagram [I've changed the 1 to e because of rendering issues I can't figure out.]Johnson He20245312025916https://hejohns.github.io/forest/hejohns-000C/hejohns-000C/forest/hejohns-000C/elementary questions, elementary toposesExpositionJohnson He20245312025916elementary questions Some basic questions, whose "elevator pitch" answers ought to roll off the tongue: What is a(n elementary) topos? What is this "topos logic" that is sound in any elementary topos? As in literally, what is the syntax? Is it the same thing as this "higher-order intuitionistic logic"? What are the inference rules/axioms? What is this "Kripke-Joyal semantics"? Is this the same thing as the so-called "forcing semantics"? What is this "Mitchell-Bénabou language"? Is it the same thing as "topos logic"? Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. Some more difficult questions we do not attempt to answer at the moment: What is a Grothendieck topos, and what's the difference v. elementary toposes? What is a "geometric logic" and a "geometric morphism"? And what's the difference v. a "logical morphism"? In what sense is "Kripke-Joyal semantics" a "forcing semantics"? What is forcing?? Why does the concept of "generic" show up everywhere? What's all this algebraic geometry stuff? And everything else… Although as you might expect, we quickly derail from "elevator pitch" answers, to "quick, write that down before I lose motivation and never complete the sentence". A truly FAQ-style answer compendium will have to come later-- after I've actually learned the basics. Johnson He20245312025916elementary answersJohnson He202461202463https://hejohns.github.io/forest/hejohns-000H/hejohns-000H/forest/hejohns-000H/Why is it called an "elementary" topos?Topoi and Computation writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2). (Recall that categories can be formalized as a two or dependently-sorted first order theory.) This is corroborated by The history of categorical logic: 1963-1977, writing that "In the years that followed, Lawvere tried to extend his analysis and sketched a categorical version of first-order theories under the name of elementary theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere introduced the notion of an elementary topos, making an explicit connection with higher-order logic and type theories. Both Lawvere and Tierney were aiming at an elementary, that is first-order, axiomatic presentation of what are now called Grothendieck toposes, a special type of categories introduced by Alexandre Grothendieck in the context of algebraic geometry and sheaf theory. Soon after, connections with intuitionistic analysis, recursive functions, completeness theorems for various logical systems, differential geometry, constructive mathematics were made." (p. 2). If Quantifiers and Sheaves is indeed the paper that introduces elementary toposes, it's curious to note that the word "elementary" never appears. It must've been coined sometime later, but I don't know the details. Johnson He202461202464https://hejohns.github.io/forest/hejohns-000G/hejohns-000G/forest/hejohns-000G/What is an elementary topos? The usual 30-second description of an elementary topos, is that "it's like " (but not quite), or "it's an axiomatization of the basic properties of ". Of course, these are always the kind of descriptions that only make sense once you've already learned what a(n elementary) topos is. We remark that The Uses and Abuses of the History of Topos Theory is precisely about this ""-based perspective. Instead, I've gathered every book I could conveniently get my hands on just now, and we'll see how each defines an elementary topos. Johnson He202461202464Topos à la Introduction to higher order categorical logic p.140definition "A topos is a cartesian closed category with a sub-object classifier and a natural numbers object." Johnson He202461202464Topos à la Sketches of an Elephant: a topos theory compendium 2.1.1definition "We are now ready to present the definition of an (elementary) topos, which is the central notion of this book: it is a category having all the good 'set-like' properties discussed in the previous chapter, except (possibly) for Booleanness. It is a remarkable fact that all these properties are implied by the conjunction of just two of them:" "A topos is a properly cartesian closed category with a subobject classifier." "… Before that, however, we must mention that the above definition of a topos, though it is appreciably simplified from that originally adopted by F. W. Lawvere and M. Tierney when they began the subject in 1969, is not the simplest possible; one can combine the exponentials and the subobject classifier into a single structure, that of power objects." ("We say a category is cartesian closed if it has finite products and all its objects are exponentiable. … Nevertheless, most of the cartesian closed categories we shall meet will have equalizers as well as finite products; we shall indicate this (with a note of mild reproof in our voice) by calling them properly cartesian closed.") (Recall that finite products equalizers finite limits.) Johnson He202461202464Topos à la Elementary Categories, Elementary Toposes 13.1definition "A topos is a Cartesian closed category with a sub-object classifier." Note though, that McLarty "assumes a base category with finite limits" when talking about cartesian closed categories. Johnson He202461202464Topos à la First Order Categorical Logicdefinition Okay so they only define a Grothendieck topos. We'll leave the discussion of Grothendieck topoi to another time. Johnson He202461202464Topos à la Category Theory 8.17definition "A topos is a category such that 1. has all finite limits, 2. has a subobject classifier, 3. has all exponentials. This compact definition proves to be amazingly rich in consequences: it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos." Johnson He202461202464Topos à la Topoi: The Categorial Analysis of Logic 4.3definition "An elementary topos is a category such that (1) is finitely complete, (2) is finitely co-complete, (3) has exponentiation, (4) has a subobject classifier. As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by (1') has a terminal object and pullbacks, and dually (2) replaced by (2') has an initial object , and pushouts. The definition just given is the one originally proposed by Lawvere and Tierney, in terms of which they started [elementary] topos theory in 1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus a topos can be defined as a Cartesian closed category with a subobject classifier. In 4.7 we shall consider a different definition, based on a categorial characterisation of power sets." (complete, meaning every diagram has a limit, finitely complete, meaning every finite diagram (finite objects, finite morphisms) has a limit, "constitute", meaning that "cartesian closed" is defined as (1) and (3), not just (3) as one expects nowadays. See, this stack overflow post.) Johnson He202461202464Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos TheoryJohnson He202461202464https://hejohns.github.io/forest/sub-000G/sub-000G/forest/sub-000G/Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48definition "[A topos is a category] with the following properties (i) has all finite limits and colimits, (ii) has exponentials, (iii) has a subobject classifier Johnson He202461202464Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161definition "A topos is a category with all finite limites, equipped with an object , with a function which assigns to each object of an object of , and, for each object of , with two isomorphisms, each natural in In other words, the functors and , the latter for each object of , are required to be representable. In the first case, the representing object is the subobject classifier, as already described in §I.3, while in the second case may be called the power object of ; moreover, the function can be extended to a functor in exactly one way so that (2) becomes natural in (as well as in ), see (7) below. Equivalently, the natural isomorphism (2) states that is the exponential , as described in §I.6; therefore, any category which satisfies (1) and has all exponentials will, in particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of §I.6. However, it will be shown in the course of this chapter that the two sets of axioms are actually equivalent. Notice that the two natural isomorphisms (1) and (2) may be combined in a single isomorphism natural in A. With , this implies (1) with , and hence also (2). In other words, we could define a topos as a category with finite limits equipped with a single operation satisfying (3)." We collect the definition of a subobject classifier, which appears in every definition, and give a standard one: 2024632024821https://hejohns.github.io/forest/hejohns-000I/hejohns-000I/forest/hejohns-000I/subobject classifierdefinition For a category , a subobject classifier is an object together with a global element such that is a pullback square. Notice that is trivially monic since every global element is trivially monic. Recall that "monos are stable under pullback". This is from Elementary Categories, Elementary Toposes. So comparing each definition with another, they're all about par for course, if we ignore natural numbers objects for now. Johnson He2024612024629https://hejohns.github.io/forest/hejohns-000F/hejohns-000F/forest/hejohns-000F/"higher-order intuitionistic logic" v. "topos logic" Many authors object to the name "higher-order intuitionistic logic". Topoi and Computation writes that "Strictly speaking, the phrase 'higher-order intuitionistic' makes little sense, since it is not clear that constructions like the power set are intuitionistically acceptable. What is meant by the phrase is a type theory built on a formally intuitionistic underlying logic." (p. 2). And Elementary Categories, Elementary Toposes similarly writes "In Part III the first order topos axioms are used to define the higher order internal language. The logic is called 'topos logic' here, although 'intuitionistic logic' is often referred to in the literature. I believe that 'intuitionism' is usually, and rightly, taken to mean Brouwer's epistemology of mathematics, which is unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in van der Hoeven and Moerdijk (1984) or many papers in Fourman et al. (1979). But topos logic coincides with no intuitionistic logic studied before toposes, and this is due to real philosophical differences (see McLarty 1990a)." (p. vii). Of course this discrepancy is what one should come to expect, since the very notion of a formalized logic of intuitionistic reasoning is at odds with Brouwer's notion of intuitionism, so when one speaks of "intuitionistic logic" it's usually in the purely formal sense. Combining the two names, I think "higher-order intuitionistic topos logic" is at least a little more descriptive. Although I like calling it "finite-type intuitionistic predicate logic" to keep in touch with some older terminology you still see around the block. 2024632025826https://hejohns.github.io/forest/hejohns-000J/hejohns-000J/forest/hejohns-000J/higher-order intuitionistic topos logic 2024632024821https://hejohns.github.io/forest/hejohns-000M/hejohns-000M/forest/hejohns-000M/finite-type intuitionistic predicate logic There is some (a posteriori insignificant) variation in the presentation of the syntax, thus following through with some input from Eric Giovannini, we juxtapose multiple definitions-- as done for elementary topoi--, but this time we slightly modify the source for uniformity as needed. 2024632024821Syntax à la Introduction to higher order categorical logic p.128definition "A type theory is a formal theory given by the following data: a class of types, including a special type a class of terms of each type, including countably many variables of each type for each finite set of variables a binary relation of entailment between terms of type all free variables of which are elements of . " 2024632024821types "The class of types is closed under the inductive clauses: , and are types ('basic' types); if and are types, so are and " 2024632024821terms "The class of terms is freely generated from certain basic terms by certain operations including the following… and is closed under certain operations. [for each type , there are countably many variables ;] is a term of type ; if is a term of type and is a term of type , then is a term of type ; if is a term of type and is a term of type , then is a term of type ; if is a term of type (possibly containing the free variable of type ), then is a term of type not containing a free occurrence of ; is a term of type ; if is a term of type so is ; and are terms of type ; if and are terms of type , so are , , ; if is a term of type (possibly containing the free variable of type ), then and are terms of type not containing free occurrences of . " 2024632024821formulas "Terms of type are also called formulas." "One may also define a number of widely used term forming operations: … , " 2024632024821Syntax à la Sketches of an Elephant: a topos theory compendium definition (We write "language" whenever Johnstone writes "signature", and use the corresponding symbol instead of . I don't know if Johnstone prefers the term "signature" because "language" is overloaded and often refers not just to the, well, language, but to the larger logic sans deductive system, but that would be a good reason.) "A language for higher-order logic is defined by specifying a set of atomic types, a set of function symbols and a set of relation symbols as in 1.1.1, together with a number of type constructors chosen from the following list, which recursively define the set of : 2024632024821types (i) Basic types: Each sort is a type (ii) Product types: There is a distinguished type (the empty product); and if and are types, then is a type. (iii) Function types: If and are types, then is a type. (iv) Power types: If is a type, the is a type. (v) List types: If is a type, then is a type. To each function symbol we assign an ordered pair of elements of ; as before, we write to indicate that has the pair of types . Similarly, each relation symbol has a type in , and we write to indicate that has type ." "If our language has the type constructors (ii) and (iv), we shall often write for the particular type . Similarly if it has constructors (ii) and (v), we write (the type of natural numbers) for ." "From now on, all our languages will contain the product type constructor; thus we have restricted all our primitive function and relation symbols to be unary. (If our signature also contaisn function types, then we can further restrict our function symbols to be constants… And if has power types, then we can replace all primitive relation symbols by constants as well)" "The term constructor for power types in fact builds terms form formulae, so over a language which contains power types we have to define terms and formulae by a simultaneous recursion". 2024632024821terms (We omit the bookkeeping of free variables that Johnstone inlines with the following definitions.) " (i) We have a stock of variables for each type . (ii) if is a function symbol of and , then . (iii) (a) we have a distinguished term . (b) If and , then . (iv) If , then and . (v) If and is a variable of type , then . (vi) If and , then . [This is formally different from (ii).] (vii) If is a formula and is a variable of type , then . (viii) For each type , there is a distinguished term of type . If and , then . (ix) The remaining term constructor associated with list types, the iterator, is more complicated [which we omit for irrelevance]. Of course, clauses (v) and (vi) of the definition above apply only if we have function types in , clause (vii) only if we have power types, and clauses (viii) and (ix) only if we have list types. Note that, except for power types, the clauses come in pairs [of constructors and destructors]. The reason why we have no 'destructor' for power types is that, just as the constructor for power types builds terms out of formulae rather than terms, the corresponding destructor builds formulae out of terms of a power type-- it thus appears as clause (iii) of the next definition." 2024632024821formulas (We again omit the bookkeeping of free variables.) "The atomic formulae over are of three kinds: (i) If is a relation symbol and , then is an atomic formulae. (ii) If and are terms of the same type , then is an atomic formula. (iii) If and for some , then is an atomic formula. The compound formulae over are built up from the atomic formulae [with ]." 2024632024821Syntax à la Elementary Categories, Elementary Toposes 14.1definition (McLarty defines the internal language of a topos directly (and its interpretation), instead of describing the logic abstractly and taking the internal language to be a diagram (in Tarskian-model theory speak) of . We rephrase McLarty's presentation to not mention a particular model . We also preserve usage of the term "sort" instead of "type".) 2024632024821types The types have the structure of the objects of an elementary topos, ie a collection of atomic types closed under . 2024632024821terms "The [language] is sorted…. We inductively define terms and their sorts: (LT1) Each sort has a list of variables over , . Every variable over is a term of sort . (LT2) For any [function symbol ] and term [of] sort , is a term of sort . [Constant symbols are also] a term of sort …. We write as a constant [of sort ]. (LT3) For every term of sort and of sort , there is a term of sort . (LT4) For every term of sort and variable of sort , there is a term of sort . " "The extension of over a list of variables is the sub-object of classified by []." 2024632024821formulas "A formula is a term of sort . By LT2 [and the existence of these terms in any elementary topos, developed in 13.3 and 13.4], for any formulas and there are formulas and .… There are also formulas , read as 'false', 'not-', and ' or ', which [with are defined in terms of ] using quantification over (15.1)." "For any formula and variable over any object , we define the formula as an abbreviation for [where is an abbreviation for , and is a term]." 2024632024821summary Ignoring the natural numbers type and the idiosyncratic list type of Sketches of an Elephant: a topos theory compendium, there is a core concept (using the notion of elementary topops for reference) of "finite-type intuitionistic predicate logic" aka "topos logic" aka "higher-order intuitionistic logic", although certain inter-definable choices can be made. There's no clear notion of "taking everything", but a reasonable "logical" choice seems to be that (over a language ,) the types are , the terms are , and the formulas are . One curiosity with taking a type and formulas to just be terms of type , is that we then have formulas like (definitionally equal to ) which don't seem very formula-like. 2024632025826semanticsJohnson He2024582025916https://hejohns.github.io/forest/hejohns-0008/hejohns-0008/forest/hejohns-0008/the core of adjoint functorsExpositionJohnson He2024572024511https://hejohns.github.io/forest/hejohns-0006/hejohns-0006/forest/hejohns-0006/the core of adjoint monotone functionslemma This is mentioned in passing, in the intro to The Core of Adjoint Functors. Johnson He2024572024511proposition Suppose and are posets, is a priori just a function, and monotone, such that . Then monotone as well. Or dually, suppose that monotone, but is a priori just a function. Then monotone as well. Johnson He2024572024511https://hejohns.github.io/forest/sub-0006/sub-0006/forest/sub-0006/proof Suppose . Notice that , by the unit of the (thin) adjunction. Then by the Hom bijection of the (thin) adjunction. (dual) Suppose . Notice that by the counit of the (thin) adjunction. Then by the Hom bijection of the (thin) adjunction. Johnson He2024572024511https://hejohns.github.io/forest/subb-0006/subb-0006/forest/subb-0006/proof corollary Notice that the above proof does not use the information that is monotone. Thus the above lemma holds where and are just a priori functions. The above lemma is in some sense the more immediately natural question, but can be easily strengthened to the form of this corollary. Johnson He2024572024821https://hejohns.github.io/forest/hejohns-0007/hejohns-0007/forest/hejohns-0007/Precursor to The Core of Adjoint Functorslemma Mentioned in passing as 2.1. Proposition., as "well-known". Johnson He2024572024821proposition Suppose is a functor, a function, and a family of natural isomorphisms. Then there is a unique extension of such that is natural in , witnessing . Or dually, suppose that is a functor, but is just a function on objects, and . Then uniquely extends to . Let's be pedantic and clarify that have the same action on morphisms as . This is conventionally left implicit, but we remark to clarify what we mean by the family of natural isomorphisms being natural. NOTE: I couldn't get よ to render in the latex (diagrams). I'm not sure if it's a lualatex/dvi or font issue. Johnson He2024572024821https://hejohns.github.io/forest/sub-0007/sub-0007/forest/sub-0007/the short proof (sketch)proof Suppose such a exists. Then naturality of requires that this square in commutes: Johnson He2024572024821https://hejohns.github.io/forest/subsub-0007/subsub-0007/forest/subsub-0007/ Then clearly . (dual) Suppose such a exists. Then naturality of requires that this square in commutes: Johnson He2024572024821https://hejohns.github.io/forest/subsubb-0007/subsubb-0007/forest/subsubb-0007/ and clearly . Johnson He20245102025927https://hejohns.github.io/forest/hejohns-000D/hejohns-000D/forest/hejohns-000D/the verbose proofproof We take inspiration from the thin case. We wts the -action on morphisms is fully determined already (∃!). That is, we have two goals: (∃) Show that there is a suitable -action. (!) That any other -action is equal to it. As usual, let denote the component of the unit of the (a posteriori) adjunction and denote the component of the counit. (∃) Now suppose . Since , we have . By the Hom bijection, . This is our candidate for the -action. . As a picture, each gives rise to the following square in : Johnson He20245102025927https://hejohns.github.io/forest/subb-000D/subb-000D/forest/subb-000D/ And our candidate is the one we know to exist by chasing around the square. We could directly try to show that defined as such is indeed functorial--and I have tried--, but it's rather difficult without the clarifying perspective of the Yoneda lemma (no, seriously, although I'm sure one could do it). It's instructive to try directly, if one hasn't. The problem is that as defined is not obviously the composition of morphisms in any diagram-- how do you reason about the functoriality of ? Instead, we want to invoke a representation theorem and work with as an explicit composition in some diagram. The solution is to follow the Yoneda principle and pass from reasoning in to . Notice that the above square is moreover natural in the variable . That is, we've actually described the component of the natural transformation . As a picture, each gives rise to the following square in : Johnson He20245102025927https://hejohns.github.io/forest/sub-000D/sub-000D/forest/sub-000D/ We first make the trivial observation that and are both Yoneda embeddings (we use the "version" where , ie the Yoneda embedding on ), and respectively. Then the Yoneda lemma tells us that by specifying an element , this fully determines (ie freely extends to) a natural transformation Equivalently, the Yoneda principle tells us that an element fully failthfully embeds as a natural transformation With this in mind, we notice that by definition, is exactly the lid to the above square. That is, both natural transformations map to the same thing-- -- by definition, thus must be equal by the Yoneda Lemma. (In hindsight, this line of reasoning is quite circumlocutious and you might as well start by saying such a would have to satisfy this square, but I think it's the more obvious starting point without presupposing it exists.) Now that we've represented as an explicit composition, we have enough to tools consider functoriality. Wts actually defines a functorial action on morphisms for a functor extending . Indeed, by definition. And suppose . Then viewed as a natural transformation gives rise to the following superimposed squares: A diagram chase verifies that the diagram indeed commutes, and that . By the ("going home" direction of the) Yoneda principle, . Thus at last, defines a functorial action on -morphisms. We should also check that is actually adjoint to -- ie, that is indeed natural in the variable as well as . By great good fortune, by defining st Johnson He2024572024821https://hejohns.github.io/forest/subsub-0007/subsub-0007/forest/subsub-0007/ it is indeed the case. (!) To finish off the result, we wts for any other extending to morphisms st , witnessed by , that . This simply follows from the observation that naturality of is exactly this square, and forces So this result is really a corollary of the Yoneda lemma. The nlab page is here. Johnson He20245102025927https://hejohns.github.io/forest/subbb-000D/subbb-000D/forest/subbb-000D/Moral Of course in hindsight, we could've (or even should've) just defined to be from the get-go, since it's a much more clearly well-behaved definition, but unless you're already sold on the Yoneda principle, I find it a difficult leap without struggling with lower-level objects a bit first. I'm still in training. This is the perspective taken in the short proof. However, unlike the thin case, we crucially use the information that is a functor so the natural transformation squares are well-defined. Johnson He20245112024821https://hejohns.github.io/forest/hejohns-000E/hejohns-000E/forest/hejohns-000E/adjunction corelemmaJohnson He20245112024821definition"An adjunction core consists of" functions and a family of isomorphisms Johnson He20245112024821proposition An adjunction core extends to an adjuction iff I commutes, and iff II commutes. The adjunction is unique when it exists. The original theorem is stated in terms of -enriched categories, but we only treat the simple case of Homset-categories here. Johnson He20245112024821https://hejohns.github.io/forest/subb-000E/subb-000E/forest/subb-000E/I Johnson He20245112024821https://hejohns.github.io/forest/subbb-000E/subbb-000E/forest/subbb-000E/II Johnson He20245112024821proof We use the same notation and definitions of as the explicit proof of the precursor lemma. The definition of in the short proof presupposes that is a functor, so we're scrupulous to avoid circularity. This proof is rather low tech and is lifted straight from the paper. It's also how you'd attack the precursor head-on. The utility is that it holds in enriched categories. The forward directions are trivial. Suppose extends to . Then I expresses that . This is exactly the condition that is natural in at , which is certainly true of an adjunction. II expresses something similar (dual). The nontrivial directions are the backward directions. (I) wts I is a functor. Suppose I holds. Then in particular, I holds at . As does this trivial square: which when pasted together, tells us magically: Combined with the earlier proof that (which just follows by definition), we've shown that defines a functorial action on morphisms. Actually, we saw earlier that I expresses that is natural in , so we've met the preconditions of the precursor lemma, thus there is a unique extension st . There's two ways ways to see the uniqueness situation: (An intermediate notion) is the unique extension of to a functor st is natural in . We've essentially seen this already, if not literally. For any functor , naturality in requires the following square to hold for all : Thus, . Then, with a functor and natural in , the precursor gives us a unique st uniquely. (Going backwards) Since the definitions of and (in the precursor) are not dependent on the other functor, the total adjunction is the unique one extending the core by applying the uniqueness result of the precusor to each functor in turn, just knowing the other is a well-defined functor. (II) wts II is a functor. This is dual. Johnson He20245112024821https://hejohns.github.io/forest/sub-000E/sub-000E/forest/sub-000E/proof corollary (For -enriched categories, ) If is thin, all diagrams commute, thus an adjunction core always determines an adjunction. isn't thin, but is, thus for posets, an adjunction core gives rise to a thin adjunction, as remarked here. I don't know how useful this lemma is for ordinary Homset categories. 2024552025927Code Contributing to Cubical Categorical Logic, "cubical agda formalizations of different approaches to categorical semantics of type theories" Gradescope-Utils, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly 2024552025927Linkshttps://mplse-reading-group.github.io/ The Second Pass of the Portable C Compiler2024552025927 Jekyll Blog 2024552025927References202410292025927https://hejohns.github.io/forest/sterling-2020/sterling-2020/forest/sterling-2020/An OK Version of Type TheoryReference202410292025927abstract What’s the right way to write down type theory (with or without universes)? It’s unclear that there is a single right way, but there are a number of good lessons we’ve learned over time that we can use to avoid certain pitfalls and bureaucratic rat-holes. These lessons include: avoid subtyping and use coercions instead, and make presupposition admissibilities hold for obvious rather than complex reasons. I haven't been able to read long articles recently, so I appreciate that this note is short and just about one or two things. AFAIU, this short note is about very syntactical matters. It's a sort of retrospective on the author's experiences with these low-level things, probably a result of the various proof assistants and type theories they'd worked on at that point. I don't feel like I 100% follow the flow, but some comments I found interesting: (single quotes are actual quotes, double quotes are air "quotes") This 'spectrum' of 'very derivable' to 'very admissible' substitution. I don't 100% understand what the author means by each approach (eg what is meant by a 'pre-term' exactly) although no doubt many people will. The 'very derivable' seems to be explicit substitutions, and can be taken to the extreme formality by only operating on the last variable in the context and requiring weakening, exchange, etc. This is not a use-facing language, but something that could be elaborated to, for example. The 'semi-admissible' seems to be the "normal" (meta?) notion of substitution on 'raw' (untyped?) terms, with the typing rules set up such that the 'typing principle of substitution' (presumably the thing that looks like ) is derivable. I should probably be ashamed that I'm not clear how one could make this derivable aside from directly throwing it in as a rule… The 'very admissible' approach seems to me, probably unfortunately, the most "normal", which is just "normal" substitution on 'raw' terms with a typing principle justified by induction on derivations. I say unfortunately because 'There is no useful consequence of the substation principle for dependent type theory being admissible but not derivable, so one might as well avoid the headache.' A similar story follows for the derivability or admissibility of presuppositions (implicit premisses) of judgments. 'Generally speaking, it is quite perverse for [] to be a rule of type theory', and although I want to agree, I suppose I don't see an obvious reason why it should be the case. I bring up this seemingly frivolous point because this seems like one way to make the presupposition derivable. Otherwise, one can make the presupposition admissible (as "normal") by having every rule premissed on its presuppositions. 'there is essentially zero pay-off to [minimizing the premiss to the rules]'. 'We do not write down any congruence rules! We assume enough rules to ensure that definitional equality is a congruence. This is one of the main motivations for working in an equational logical framework (to rule out any perverse kind of “equality” that is not a congruence!)' 'But let me state what is not negotiable: the most well-adapted notion of substitution for dependent type theory is a simultaneous substitution [inline figure omitted]. So, a definition of when such a simultaneous substitution is well-typed should be given (either by rules, or by induction on contexts)' And lastly, that there are many ways to handle universes and rules for them, and that some 'are reasonable but even harder to justify in many models', or that some 'has many more useful models'. There's a line of work on what this means precisely, which I should get to sooner than later. paper 2024612025927https://hejohns.github.io/forest/mclarty-1990a/mclarty-1990a/forest/mclarty-1990a/Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)Reference paper 20245132025927https://hejohns.github.io/forest/scherer-2017/scherer-2017/forest/scherer-2017/Deciding equivalence with sums and the empty typeReferencehttps://arxiv.org/pdf/1610.01213. The nice introduction informed me of Equality Between Functionals and Equality Between Functionals Revisited. Referenced on stack overflow. 20245132025927https://hejohns.github.io/forest/friedman-1975/friedman-1975/forest/friedman-1975/Equality Between FunctionalsReference20245132025927abstract The λ-calculus, in both its typed and untyped forms, has primarily been regarded as an attempt to formalize the concept of rule or process, and ultimately to provide a new foundation for mathematics. It seems fair to say that this aspect of the λ-calculus is currently in an embryonic state of development, awaiting further conceptual advances. There is, however, another aspect of the typed λ-calculus, which is readily understood. This is its connection with the full classical finite type structure over ω, (i.e., with the functionals of finite type). In an obvious way, each closed term in the typed λ-calculus defines a functional of finite type over ω. Call a functional simple if it is given by some closed term in the typed x-calculus. Several definitions of convertibility between terms in the λ-calculus, with or without types, have been considered. The motivation for introducing these definitions has primarily been to analyze the notion of the identity between rules, or processes. The relation defined in the text, is equivalent to one of these definitions of convertibility. It is easy to see that any two convertible terms define the same functional of finite type. We show here that any two non- convertible terms define different functionals of finite type.[This result was obtained in 1970.] This, coupled with the known decidability of convertibility, tells us that "equality between simple functionals is recursive." Square brackets denote author's footnotes. paper 20245132025927https://hejohns.github.io/forest/statman-1985/statman-1985/forest/statman-1985/Equality Between Functionals RevisitedReference20245132025927abstract In this note we shall try to place Friedman's remarkable little paper (3) in the context of what we know today about the model theory of the typed λ-calculus. In order to do this, it is appropriate to survey recent work in this area, in so far as it touches on issues raised by Friedman. We don't intend a general survey; much will be left out. In particular, we shall omit discussion of unification, fragments of L.C.F., and polymorphic types, which are areas of interest to the author, and the specialist will surely find other omissions. However, we will try to show the reader how Friedman's paper lays the foundation for the general model theory of the typed λ-calculus. The plan of this note is the following. We shall begin by a brief, informal introduction to the subject. The reader is then advised to read Friedman's paper. It is, after all, quite accessible and easy to read. We shall then proceed to consider the three major issues touched on in "Equality between functionals" as we view them today. These issues are completness theorems, the solvability of higher type functional equations, and logical relations. paper 20245242025927https://hejohns.github.io/forest/sterling-2022/sterling-2022/forest/sterling-2022/Naïve Logical Relations in Synthetic Tait ComputabilityReference20245242025927abstract Logical relations are the main tool for proving positive properties of logics, type theories, and programming languages: canonicity, decidability, conservativity, computational adequacy, and more. Logical relations combine pure syntax with non-syntactic objects that are parameretized in syntax in a somewhat complex way; the sophistication of possible parameterizations makes logical relations a tool that is primarily accessible to specialists. In the spirit of Halmos’ Naïve Set Theory, we advance a new viewpoint on logical relations based on synthetic Tait computability, an internal language for categories of logical relations. In synthetic Tait computability, logical relations are manipulated as if they were sets, making the essence of many complex logical relations arguments accessible to non-specialists. 2024524202592720245252025927 I can't quite follow the phase distinction and only have a very vague idea of what's going on starting at p. 9. From what I can tell, the idea is to work in the internal language of some topos-- really the internal "agda-like dependent type theory"-- to construct the logical relations category, instead of doing something like (analytic) gluing. How this makes anything easier is beyond me at the moment. I can't tell if the object-space/meta-space discussion is clear and I just don't have some context, or it's actually presented almost mystically. The mention of finally tagless encodings and a justification for hoas doesn't seem super on-the-point, but if it's mentioned it must be for good reason… Again, quotes are air-quotes, not actual quotes. paper 202411112025927https://hejohns.github.io/forest/Grunbacher-2003/Grunbacher-2003/forest/Grunbacher-2003/POSIX Access Control Lists on LinuxReference202411112025927abstract This paper discusses file system Access Control Lists as implemented in several UNIX-like operating systems. After recapitulating the concepts of these Access Control Lists that never formally became a POSIX standard, we focus on the different aspects of implementation and use on Linux. paper 2024612025927https://hejohns.github.io/forest/lawvere-1970/lawvere-1970/forest/lawvere-1970/Quantifiers and SheavesReference2024612025927abstract The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the present joint work with Myles Tierney there are important influences in the other direction: a Grothendieck "topology" appears most naturally as a modal operator, of the nature "it is locally the case that", the usual logical operators such as ∀, ∃, ⇒ have natural analogues which apply to families of geometrical objects rather than to propositional functions, and an important technique is to lift constructions first understood for "the" category S of abstract sets to an arbitrary topos. We first sum up the principal contradictions of the Grothendieck-Giraud-Verdier theory of topos in terms of four or five adjoint functors, significantly generalizing the theory to free it of reliance on an external notion of infinite limit (in particular enabling one to claim that in a sense logic is a special case of geometry). The method thus developing is then applied to intrinsically define the concept of Boolean-valued model for S (BVM/S) and to prove the independence of the continuum hypothesis free of any use of transfinite induction. The second application of the method outlined here is an intrinsic geometric construction of the Chevalley-Hakim global spectrum of a ringed topos free of any choice of a "site of definition". When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing. Doing this for "set theory" requires taking account of the experience that the main pairs of opposing tendencies in mathematics take the form of adjoint functors, and frees us of the mathematically irrelevant traces () left behind by the process of accumulating () the power set () at each stage of a metaphysical "construction". Further, experience with sheaves, permutation representations, algebraic spaces, etc., shows that a "set theory" for geometry should apply not only to abstract sets divorced from time, space, ring of definition, etc., but also to more general sets which do in fact develop along such parameters. For such sets, usually logic is "intuitionistic" (in its formal properties) usually the axiom of choice is false, and usually a set is not determined by its points defined over only. I have not read this, and have no idea what "contradictions" he's referring to. paper 2024612025927https://hejohns.github.io/forest/blass-2003/blass-2003/forest/blass-2003/Resource Consciousness in Classical LogicReference2024612025927abstract Using Herbrand's Theorem, we define simple Herbrand validity, a sort of resource consciousness that makes sense in classical predicate logic. We characterize the propositional formulas all of whose first-order instances are simply Herbrand valid. The characterization turns out to coincide with a known characterization of game semantical validity for multiplicative formulas. This is currently a "would like to read, but no urgency" paper, because Blass + Herbrand's theorem. paper 20259272025927https://hejohns.github.io/forest/hofmann-1999/hofmann-1999/forest/hofmann-1999/Semantical analysis of higher-order abstract syntaxReference20259272025927abstract A functor category semantics for higher-order abstract syntax is proposed with the following aims: relating higher-order and first order syntax, justifying induction principles, suggesting new logical principles to reason about higher-order syntax. paper 2024612025927https://hejohns.github.io/forest/mclarty-1990b/mclarty-1990b/forest/mclarty-1990b/The Uses and Abuses of the History of Topos TheoryReference2024612025927abstract The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with remarks on a categorical view of the history of set theory, including a false history plausible from that point of view that would make it helpful to introduce toposes as a generalization from set theory. paper 2024612025927https://hejohns.github.io/forest/marquis-reyes-2012/marquis-reyes-2012/forest/marquis-reyes-2012/The history of categorical logic: 1963-1977Reference I have not read this. This might be a good source, aside from Ronnie Chen, where the viewpoint of "generalized algebraic semantics" is developed. paper 2024612025927https://hejohns.github.io/forest/blass-1988/blass-1988/forest/blass-1988/Topoi and ComputationReference2024612025927abstract This is the written version of a talk given at the C.I.R.M. Workshop on Logic in Computer Science at Marseille, France in June, 1988. Its purpose is to argue that there is a close connection between the theory of computation and the geometric side of topos theory. We begin with a brief outline of the history and basic concepts of topos theory. 20246120259272024612025927notes Prof. Blass essentially described this background over the course of the logic seminar last semester (as comments at various meetings, culminating with a full talk), but this is a nice written account of the history of topos theory, the relation between Grothendieck and Elementary topoi, and the relation between geometric and logical morphisms. paper Johnson He20245282025927https://hejohns.github.io/forest/quine-1960/quine-1960/forest/quine-1960/Variables Explained AwayReference "Separately I sent a copy of Quine's 1950 paper on combinatory logic to this list. Whatever else you might think about Quine, he was a terrific explainer, and this is the best introduction to the subject I know." (Yes, that 50 should be a 60(b).) Here is the followup I wrote: Johnson He20245282025927email Thanks for sharing this nice little note, which sketches a combinatory (classical) first-order logic. I was surprised by the mention of von Neumann 1925 in this snippet: "In a way our operators are reminiscent also of the axioms of class existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre, Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite dissimilarity of purpose." (Footnote 1, p. 346) From what I gather, the reminiscence with von Neumann 1925 (I assume Quine was principally referring to the "III. Logical Construction Axioms") is that the purposes (of Quine's predicate combinators and von Neumann's axioms of existence of II-objects) are actually quite similar. It seems that the system of von Neumann 1925 isn't couched in any particular formal system (although it's maybe implicitly first-order, and apparently translated into such by Skolem 1938), thus von Neumann wants to recover from any definite property ("logical condition") on I-objects, the (existence of a) class (II-object) of those I-objects. Wikipedia calls this the class existence theorem. (More generally, the "reduction theorem" in von Neumann 1925 says that every expression of I-objects (definable class-function) is indeed a II-object (class-function). This restricts to logical conditions and classes.) To do this economically, the existence of a few basic combinator-classes are stipulated (=, ∀, ∃!), and more complex logical conditions are put in "normal form" by eliminating variables with the combinator-classes. The 'uneconomic' way, if von Neumann 1925 were working in an explicit logic, would be to just stipulate a schema of class comprehension (also mentioned in the Wikipedia article), as included in the back of Chang and Keisler. Actually, while there are explicit class existence axioms for absorbing =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how they'd follow with the other axioms, but I suppose they must. The Wikipedia article has a write-up starting from more user-friendly axioms. In any case, I hadn't realized that the historic move to make precise the notion of 'definite property' as 'first-order definable' was nontrivial, and found this connection with Quine's note interesting, Johnson He paper An Axiomatization of Set Theory20245212025927https://hejohns.github.io/forest/maddy-2019/maddy-2019/forest/maddy-2019/What do we want a foundation to do?Reference20245212025927abstract Mainstream orthodoxy holds that set theory provides a foundation for contemporary pure mathematics. Critics of this view argue that category theory, or more recently univalent foundations, is better suited to this role. Some observers of this controversy suggest that it might be resolved by a better understanding of what a foundation is. Despite considerable sympathy to this line of thought, I’m skeptical of the unspoken assumption that there’s an underlying concept of a ‘foundation’ up for analysis, that this analysis would properly guide our assessment of the various candidates. In contrast, it seems to me that the considerations the combatants offer against opponents and for their preferred candidates, as well as the roles each candidate actually or potentially succeeds in playing, reveal quite a number of different jobs that mathematicians want done. What matters is these jobs we want our theories to do and how well they do them. Whether any or all of them, jobs or theories, deserves to be called ‘foundational’ is really beside the point. A nice article that can be briefly summarized as There is no one clear set of desiderata for a foundational system, whatever that means Classical set theory is perfectly fine as a foundational system in the traditionally (ie ZFC) conceived sense There is promise for the new 21ˢᵗ century desiderata of proof checking (recalling Voevodsky's concerns with correctness "at the bleeding edge"). The contrast between the minimalistic ZFC and potentially enormous "user friendly" univalent type theories is reminiscent of Hilbert style proofs vs natural deduction. Indeed, it's like how Ronnie calls set theory "the machine/assembly language of math" (kind of paraphrase), and Max calls SK combinators "the machine/assembly language of λ-calculus". Natural deduction is certainly more complicated than SK/Hilbert proof systems, but has nice usability properties that are worth study and implementation. The general mathematical trend seems to be that of starting with a spartan formal system, then gradually making it more user friendly to accomodate every increasingly complicated mathematics, where the minimalistic system starts to leave one wanting. Maddy only mentions Coq, but being 2019 it would've been nice if somewhat redundant to mention other proof assistants eg Cubical Agda. I've used quotes here as air quotes, not actual quotes of Maddy. paper 20241122025916https://hejohns.github.io/forest/Adámek-Rosický-Vitale-2010/Adámek-Rosický-Vitale-2010/forest/Adámek-Rosický-Vitale-2010/Algebraic Theories : A Categorical Introduction to General AlgebraReference20245282025916https://hejohns.github.io/forest/von-neumann-1925/von-neumann-1925/forest/von-neumann-1925/An Axiomatization of Set TheoryReference One of the early NBG papers. Maybe "the" NBG paper. My notes on this are currently in Variables Explained Away. English translation in From Frege To Gödel. 20247112025916https://hejohns.github.io/forest/grayson-2018/grayson-2018/forest/grayson-2018/An introduction to univalent foundations for mathematiciansReference"An gentle introduction to univalent foundations.", but still made much easier to read after the maxsnew/cubical-categorical-logic hacking experience. The "two examples, punchline third" expository pattern is very nice. https://arxiv.org/abs/1711.01477v32024562025916https://hejohns.github.io/forest/jacobs-1999/jacobs-1999/forest/jacobs-1999/Categorical Logic and Type TheoryReference2024622025916https://hejohns.github.io/forest/awodey-2010/awodey-2010/forest/awodey-2010/Category TheoryReference First book I used to seriously study category theory. Terse. No frills. A little about logic. Good. 20245312025916https://hejohns.github.io/forest/mclarty-1992/mclarty-1992/forest/mclarty-1992/Elementary Categories, Elementary ToposesReference20245242025916https://hejohns.github.io/forest/carette-kiselyov-shan-2007/carette-kiselyov-shan-2007/forest/carette-kiselyov-shan-2007/Finally Tagless, Partially EvaluatedReference20245242025916abstract We have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or ML) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value CPS transformers. Our main idea is to encode HOAS using cogen functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the λ-calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and CPS transformations. Our encoding of an object term abstracts over the various ways to interpret it, yet statically assures that the interpreters never get stuck. To achieve self-interpretation and show Jones-optimality, we relate this exemplar of higher-rank and higher-kind polymorphism to plugging a term into a context of let-polymorphic bindings. 2024524202591620245252025916notes This is the APLAS version (conference paper). There is a roughly twice as long journal version. I didn't study a lot of the probably more interesting aspects of this work. I just wanted a first/resonant-understanding of "what does finally tagless mean?". It seems the very basic idea is about the general problem of DSLs. The very short way to say it, is that we like to construct languages as object languages of nice meta languages. For example, a nice meta languages gives us nice primitives to reuse as the denotational semantics of the object language. A classic example is to use the Int type in the meta language as the semantics of the DSL/embedded/object language's numeric type. Most people's knee jerk reaction is to encode the object language as an ADT. This is fine and well, and can be called initial in the sense that it is the free syntax generated by the encoded rules/initial algebra/inductive dataype. But the naïve TODO 2024622025916https://hejohns.github.io/forest/makkai-reyes-1977/makkai-reyes-1977/forest/makkai-reyes-1977/First Order Categorical LogicReference I've heard/been told this is the book for, well, the subject in the title, and Makkai completeness. 20245282025916https://hejohns.github.io/forest/van-heijenoort-1967/van-heijenoort-1967/forest/van-heijenoort-1967/From Frege To GödelReference A+. I should get a second copy to gift whenever needed. Johnson He20245112025916https://hejohns.github.io/forest/millies-zeilberger-2015/millies-zeilberger-2015/forest/millies-zeilberger-2015/Functors are Type Refinement SystemsReference20245112025916abstract The standard reading of type theory through the lens of category theory is based on the idea of viewing a type system as a category of well-typed terms. We propose a basic revision of this reading: rather than interpreting type systems as categories, we describe them as functors from a category of typing derivations to a category of underlying terms. Then, turning this around, we explain how in fact any functor gives rise to a generalized type system, with an abstract notion of typing judgment, typing derivations and typing rules. This leads to a purely categorical reformulation of various natural classes of type systems as natural classes of functors. The main purpose of this paper is to describe the general framework (which can also be seen as providing a categorical analysis of refinement types), and to present a few applications. As a larger case study, we revisit Reynolds’ paper on “The Meaning of Types” (2000), showing how the paper’s main results may be reconstructed along these lines. Johnson He20245112025916Sections 1-2What is a logical relation? A LR is just a model of the syntax, used to prove statements. Thus, it's usually just a model "above" (or a structure preserving functor into) the syntax, typically trivially constructed as a pullback (it's hard in general to concoct models with structure preserving maps into the syntax). This is so we can take advantage of the universal property of the syntax (its initiality), as we always do (eg classical soundness of Prop logic). The initiality lets us lift any syntactic term into the model, and talk about it in the model. The projection of the model then let's us actually say something about the syntax itself. So a logical relations argument is just a clever choice of model, to let us say something interesting/talk about what we care about. So (classical) parametricity (a la Reynolds '83) is just this. He constructs a set-theoretic model of system F (all models in the paper are set theoretic) out of two standard models. In terms of data abstraction/automata, the point is to construct a bisimulation between the two models, so we can say that two types are "the same"-- every step in the program preserves the relation in lock-step. This is entirely analogous to canonicity, where we need to actually reason about the whole language preserving the type, not just intro and elims for that type, for a stronger inductive hypothesis. This way, we can show that two different semantics and implementations for a intset type are in a precise sense, "the same". We typically relate a naive implementation and a optimized implementation. That is, the reference and optimized implementations are bisimilar/extensionally equivalent. I'm still not super clear on why parametricity fundamentally is tied with this notion of data hiding. I guess it's that to give an account of "operating on a type, just knowing it's a type", one has to give a notion of "just knowing it's a type". For this, Reynolds defines a notion of "respecting interfaces" w/ relations between models, and that polymophic terms respect any interface? That last question is that the "constructing a bisimulation between two models" view seems to make sense, but as a phrasing of parametricity, it should really be that "for any two models, we can always construct (using a LR) a bisimulation between them". Assuming it's actually true eg you can't construct the model if you say that the empty intlist is related to a nonempty one. And do we get anything interesting if one of the models is itself the syntax? Is this what parametricity is? What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate. What about the syntax as a "closed terms" henkin model? Maybe I'm finally close to being able to understand what a Kripke logical relation is. 2024632025916https://hejohns.github.io/forest/hinman-2005/hinman-2005/forest/hinman-2005/Fundamentals of Mathematical LogicReference The tome, sometimes used at Michigan, but it was never (one of) my course text(s). I haven't used it much for reading either, owing to the large amount of new notation one has to refresh every time you open the book, but it's a solid reference book. 2024632025916https://hejohns.github.io/forest/lambek-scott-1986/lambek-scott-1986/forest/lambek-scott-1986/Introduction to higher order categorical logicReference20246152025916https://hejohns.github.io/forest/shoenfield-1967/shoenfield-1967/forest/shoenfield-1967/Mathematical LogicReference20246152025916https://hejohns.github.io/forest/logicmatters/logicmatters/forest/logicmatters/Peter Smith's Book ReviewsReference One of my all-time favorite pages on the web (and that's not an uncommonly held opinion), a tremendous resource and service for the community. I'm not able to evaluate too many of Prof. Smith's opinions, but I don't disagree with any of what he's written about the books I have seen. At least I just reread the review of Mathematical Logic, and "So this neither has the cleanness of a Hilbert system nor the naturalness of a natural deduction system. As far as I noticed, nothing is said to motivate this seemingly horrible choice as against others." always makes me laugh. https://www.logicmatters.net/tyl/booknotes/20246302025916https://hejohns.github.io/forest/rosiak-2022/rosiak-2022/forest/rosiak-2022/Sheaf Theory through ExamplesReference open access 2024622025916https://hejohns.github.io/forest/maclane-moerdijk-1994/maclane-moerdijk-1994/forest/maclane-moerdijk-1994/Sheaves in Geometry and Logic: A First Introduction to Topos TheoryReference2024622025916https://hejohns.github.io/forest/johnstone-2002/johnstone-2002/forest/johnstone-2002/Sketches of an Elephant: a topos theory compendiumReference2024572025916https://hejohns.github.io/forest/street-2012/street-2012/forest/street-2012/The Core of Adjoint FunctorsReference The first point: 2024572025916abstract During my Topology lectures at Macquarie University in the 1970s, the students and I realized, in proving that a function f between posets was order preserving when there was a function u in the reverse direction such that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order preserving. I realized then that knowing functors in the two directions only on objects and the usual hom adjointness isomorphism implied the effect of the functors on homs was uniquely determined. Writing this down properly led to the present paper. Johnson He20245122025916https://hejohns.github.io/forest/dummett-1991/dummett-1991/forest/dummett-1991/The Logical Basis of MetaphysicsReferenceJohnson He20245122025916Stability The opening sentence is what Noam Zeilberger quotes on his work on polarity. What's the relationship between harmony-stability and Pfenning's local soundness and completeness? They're not literally the same thing, but they're similar. 202411242025916https://hejohns.github.io/forest/scott-1980/scott-1980/forest/scott-1980/The Presheaf Model for Set TheoryReference (some paper notes TODO) 2024622025916https://hejohns.github.io/forest/goldblatt-2006/goldblatt-2006/forest/goldblatt-2006/Topoi: The Categorial Analysis of LogicReference "Bibliographical Note: This Dover edition, first published in 2006, is a slightly corrected unabridged republication of the revised (second) edition of the work originally published in 1984 by Elsevier Science Publishers B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series, "Studies in Logic and the Foundations of Mathematics." A new Preface and updated Bibliography have been specially prepared for this reprint." References Context Backlinks Related Johnson He 2024 5 5 2025 9 27 https://hejohns.github.io/forest/hejohns/ hejohns /forest/hejohns/ Johnson He person Johnson He20246102025927https://hejohns.github.io/forest/hejohns-000O/hejohns-000O/forest/hejohns-000O/About Me I am a PhD student in Computer Science at Indiana University. My advisor is Carlo Angiuli. I'm interested in logic and related areas of computer science, particularly programming languages. This is my forest. email: photo: Johnson He2024552025927Extended About Previously, I was a master's student (SUGS) in Computer Science and Engineering at the University of Michigan, where I was an undergraduate before that. email (active indefinitely): My screen name is hejohns, eg https://github.com/hejohns, but IU only let you choose from a couple automatically generated options, so my username is hejo there. Contributions