 Oct 2024


Baez, John C., and Mike Stay. “Physics, Topology, Logic and Computation: A Rosetta Stone.” Quantum Physics; Category Theory. arXiv:0903.0340 [QuantPh] 813 (March 2, 2009): 95–172. https://doi.org/10.1007/9783642128219_2.

 Dec 2023

climateuncensored.com climateuncensored.com

the wealthiest 1% of people on the planet are responsible for double the greenhouse gas emissions of the poorest half

for: carbon inequality, question  new COP  focused on elites?

comment
 while COP28 fights over which nations bear what responsibility, from this perspective, there is an entirely different class of people that must be held responsible, not at the nation state level, but at the individual level. Why isn't there a COP where the elites are held responsible?

question
 Are we making a grave category error in holding the wrong class of people responsible? Should questions of carbon equity concern both high polluting nations AND individuals?
 At the very least, should we formally recognize a parallel set of responsibilities and elevate that recognition to the level of COP conventions to deal with the problem?



cuissmalltalk.github.io cuissmalltalk.github.io

How many methods are there in the arithmetic method category of the CharacterSequence class?
There are no messages categorized as arithmetic in String but in CharacterSequence, which String subclassifies. Squeak Strings on the other hand, implement arithmetic messages, exactly those mentioned in the solution to the exercise; Note however that
String allMethodsInCategory: #arithmetic
in both Cuis and Squeak include a seventh method#raisedTo:
because#allMethodsInCategory
also looks up for inherited methods from superclasses. Yes! you can raise the empty string to the power of n for n ∈ Number and the result is another empty string! WAT :) The fact that we can evaluate'' raisedTo: n
without error seems like an unforeseen consequence of#adaptToCollection:andSend:
relying on#collect
and how the latter works on empty collections.

 Sep 2023
 Aug 2023


category theory + zettelkasten... hmmm... feels a bit like Leibniz chasing a universal language
what is a zettelkasten monad?


hub.jhu.edu hub.jhu.edu

In the ensuing decades, mathematicians began working with this new thing, the category, and this new idea, equivalence. In so doing they created a revolutionary new approach to mathematics, category theory, that many see as supplanting set theory. Imagine if writers had spent 150 years representing the world only through basic description: This is a red ball. That is a 60foot tree. This is a dog. Then one day someone discovers metaphor. Suddenly, our ability to find new ways to represent the world explodes, as does our knowledge of writing as a discipline.
I love the idea here of analogizing the abstract nature of category theory in math with the abstract nature of metaphor in writing!
Good job Dale Keiger!


"If mathematics is the science of analogy, the study of patterns, then category theory is the study of patterns of mathematical thought."
original source?

 Jun 2023

docdrop.org docdrop.org

The intermediary category contains three modes: Dorian, Locrian, and Locrian Ω2

 May 2023

docdrop.org docdrop.org

with their new different and perhaps bigger brains the AIS of the future may prove themselves to be better adapted to 00:19:05 life in this transhuman world that we're in now
 comment
 Is this not a category error in classifying inert technology as life?
 When does an abiotic human cultural artefact become a living form?
 comment

 Apr 2023

forum.zettelkasten.de forum.zettelkasten.de

[Zettel feedback] Functor (Yeah, just that)
reply to ctietze at https://forum.zettelkasten.de/discussion/2560/zettelfeedbackfunctoryeahjustthat#latest
Kudos on tackling the subject area, especially on your own. I know from experience it's not as straightforward as it could/should be. I'll refrain from monkeying with the perspective/framing you're coming from with overly dense specifics. As an abstract mathematician I'd break this up into smaller pieces, but for your programming perspective, I can appreciate why you don't.
If you want to delve more deeply into the category theory space but without a graduate level understanding of multiple various areas of abstract mathematics, I'd recommend the following two books which come at the mathematics from a mathematician's viewpoint, but are reasonably easy/intuitive enough for a generalist or a nonmathematician coming at things from a programming perspective (particularly compared to most of the rest of what's on the market):
 Ash, Robert B. A Primer of Abstract Mathematics. 1st ed. Classroom Resource Materials. Washington, D.C.: The Mathematical Association of America, 1998.
 primarily chapter 1, but the rest of the book is a great primer/bridge to higher abstract math in general)
 Spivak, David I. Category Theory for the Sciences. MIT Press, 2014.
 Spivak had a free draft prepublication version at https://arxiv.org/abs/1302.6946; There's also an open courseware version of his MIT course with a free .pdf textbook (with a 2013 date, but ostensibly a copy of the final published version from 2014) at https://ocw.mit.edu/courses/18s996categorytheoryforscientistsspring2013/
You'll have to dig around a bit more for them (his website, Twitter threads, etc.), but John Carlos Baez is an excellent expositor of some basic pieces of category theory.
For an interesting framing from a completely nontechnical perspective/conceptualization, a friend of mine wrote this short article on category theorist Emily Riehl which may help those approaching the area for the first time: https://hub.jhu.edu/magazine/2021/winter/emilyriehlcategorytheory/?ref=dalekeiger.net
One of the things which makes Category Theory difficult for many is that to have multiple, practical/workable (homework or inbook) examples to toy with requires having a reasonably strong grasp of 34 or more other areas of mathematics at the graduate level. When reading category theory books, you need to develop the ability to (for example) focus on the algebra examples you might understand while skipping over the analysis, topology, or Lie groups examples you don't (yet) have the experience to plow through. Giving yourself explicit permission to skip the examples you have no clue about will help you get much further much faster.
I haven't maintained it since, but here's a site where I aggregated some category theory resources back in 2015 for some related work I was doing at the time: https://cat.boffosocko.com/courseresources/ I was aiming for basic/beginner resources, but there are likely to be some highly technical ones interspersed as well.
 Ash, Robert B. A Primer of Abstract Mathematics. 1st ed. Classroom Resource Materials. Washington, D.C.: The Mathematical Association of America, 1998.


bartoszmilewski.com bartoszmilewski.com

Here we have extended this model to a slightly different category, a category where morphisms are represented by embellished functions, and their composition does more than just pass the output of one function to the input of another. We have one more degree of freedom to play with: the composition itself. It turns out that this is exactly the degree of freedom which makes it possible to give simple denotational semantics to programs that in imperative languages are traditionally implemented using side effects.

For our limited purposes, a Kleisli category has, as objects, the types of the underlying programming language. Morphisms from type A to type B are functions that go from A to a type derived from B using the particular embellishment. Each Kleisli category defines its own way of composing such morphisms, as well as the identity morphisms with respect to that composition.

 Feb 2023

bartoszmilewski.com bartoszmilewski.com

There is just one little nit for mathematicians to pick: morphisms don’t have to form a set. In the world of categories there are things larger than sets. A category in which morphisms between any two objects form a set is called locally small.

It’s worth emphasizing that Haskell lets you express equality of functions, as in: mappend = (++) Conceptually, this is different than expressing the equality of values produced by functions, as in: mappend s1 s2 = (++) s1 s2 The former translates into equality of morphisms in the category Hask (or Set, if we ignore bottoms, which is the name for neverending calculations). Such equations are not only more succinct, but can often be generalized to other categories. The latter is called extensional equality, and states the fact that for any two input strings, the outputs of mappend and (++) are the same. Since the values of arguments are sometimes called points (as in: the value of f at point x), this is called pointwise equality. Function equality without specifying the arguments is described as pointfree. (Incidentally, pointfree equations often involve composition of functions, which is symbolized by a point, so this might be a little confusing to the beginner.)

Is composition associative? Check!
Associativity for
<=
seems a bit weird to me; generally you want the value inside of()
to be comparable outside; but(1 <= 2) <= 3
would result inTrue <= 3
. So I think the best way to think of associativity here is that it just does not apply, and the()
can be dropped. 
Let’s characterize these ordered sets as categories. A preorder is a category where there is at most one morphism going from any object a to any object b. Another name for such a category is “thin.” A preorder is a thin category.

A set of morphisms from object a to object b in a category C is called a homset and is written as C(a, b) (or, sometimes, HomC(a, b)). So every homset in a preorder is either empty or a singleton. That includes the homset C(a, a), the set of morphisms from a to a, which must be a singleton, containing only the identity, in any preorder. You may, however, have cycles in a preorder. Cycles are forbidden in a partial order.


www.reddit.com www.reddit.com

Chris, Chris... concepts and propositions are not nebulous dictionary definitions unless you're joining the Frankfurt School :) :).Click here to get some clarity about these basic terms as applied to learning: https://cmap.ihmc.us/docs/concept.phpAbout systems and emergence, I prefer Mario Bunge's book: https://www.amazon.com/EmergenceConvergenceQualitativeKnowledgePhilosophy/dp/1442628219Irony? No way. You are always bringing new information about the historical roots of Zettelkasten. Keep doing that, please! Thanks!
reply to u/NewInvestigator623 at https://www.reddit.com/r/antinet/comments/10r6uwp/comment/j784srg/?utm_source=reddit&utm_medium=web2x&context=3
I meant nebulous for my initial purposes. They obviously have very concrete meanings in more specific contexts, though even there they can vary. I saw your other post on concept maps where I imagine they matter more; some of that reminds me about some of my initial explorations into category theory (math) a few years back. I'm curious what the overlap of those two looks like...
On systems, complexity, and emergence, I'm probably closer to the school of thought and applications coming out of the Santa Fe Institute. I'll have to look at Bunge's work there, I've only glanced at some of his math/physics work but never delved into his philosophical material.

 Jan 2023

arxiv.org arxiv.org

Hypergraph Categories
[[hypergraph categories]]: https://anagora.org/hypergraph+categories
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shopify.engineering shopify.engineering

Of all the programming languages that are used at Shopify, Ruby is the one that most developers are familiar with, followed by Node, Go, and Rust.
Node is not a programming language.

 Dec 2022

www.logicmatters.net www.logicmatters.net

for settling in a finite number of steps, whether a relevant object hasproperty P.Relatedly, the answer to a question Q is effectively decidable ifand only if there is an algorithm which gives the answer, again by adeterministic computation, in a finite number of steps.
Missing highlight from preceding page:
A property \( P \) is effectively decidible if and only if there is an algorithm (a finite set of instructions for a deterministic computation) ...
Isn't this related to the idea of left & right adjoints in category theory? iirc, there was something about the "canonical construction" of something X being the best solution to a particular problem Y (which had another framing like, "Problem Y is the most difficult problem for which X is a solution")
Different thought: the CurryHowardLambek correspondance connects intuitionistic logic, typed lambda calculus, and cartesian closed categories.

 Nov 2022

docdrop.org docdrop.org

okay so remind you what is a sheath so a sheep is something that allows me to 00:05:37 translate between physical sources or physical realms of data and physical regions so these are various 00:05:49 open sets or translation between them by taking a look at restrictions overlaps 00:06:02 and then inferring
Fixed typos in transcript:
Just generally speaking, what can I do with this sheaftheoretic data structure that I've got? Okay, [I'll] remind you what is a sheaf. A sheaf is something that allows me to translate between physical sources or physical realms of data [in the left diagram] and the data that are associated with those physical regions [in the right diagram]
So these [on the left] are various open sets [an example being] simplices in a [simplicial complex which is an example of a] topological space.
And these [on the right] are the data spaces and I'm able to make some translation between [the left and the right diagrams] by taking a look at restrictions of overlaps [a on the left] and inferring back to the union.
So that's what a sheaf is [regarding data structures]. It's something that allows me to make an inference, an inferential machine.


docdrop.org docdrop.org

i think so like in social terms the conservatives would say well i like that it benefits from the wisdom of math already invented you're not 00:36:39 throwing anything away you're not you're not throwing it all away and starting over you're taking what we already have and you're you're using it that's great and a libertarian might say i really like that you're free to create as you see fit you can make anything you 00:36:52 want and you're working within this background framework that's minimally invasive it doesn't make a lot of rules for you but it is highly functional i like that it kind of keeps everyone in line while 00:37:03 like satisfying some formal contracts or something while still being uh i'm still free to create and a progressive might say i like about category that theory that everyone can contribute to 00:37:15 making their own world making it more rich adding new ideas uh making it more meaningful understanding connections between things a modern viewpoint would say i like that 00:37:26 it's completely rigorous that it's been used in proving wellknown conjectures that people thought were important to prove but also that it's interesting it's useful in science and technology and a postmodern person might say i like 00:37:40 that um that no perspective is right that that there's just all sorts of different categories but that navigating between these perspectives lets you look at problems from all sides or a hippie might say i like that it's 00:37:53 all about relationship and connection or irrelevant i don't know what that means maybe a practical person might say that i like that it's that we can actually use it to organize and learn from big data in 00:38:06 today's world or to manage complexity of software projects that are that are very large and changing all the time i like that you can think about ai and other complex systems with this stuff i think it's relevant and 00:38:19 practical for right now so that's that's my uh tutorial or that's the the part i'm going to record and now i'm going to open it up for questions
David Spivak discusses how category theory may appeal to different political ideologies for a variety of reasons.

 Oct 2022

www.loom.com www.loom.com

https://www.loom.com/share/a05f636661cb41628b9cb7061bd749ae
Synopsis: Maggie Delano looks at some of the affordances supplied by Tana (compared to Roam Research) in terms of providing better blockbased user interface for note type creation, search, and filtering.
These sorts of tools and programmable note implementations remind me of Beatrice Webb's idea of scientific note taking or using her note cards like a database to sort and search for data to analyze it and create new results and insight.
It would seem that many of these note taking tools like Roam and Tana are using blocks and sub blocks as a means of defining atomic notes or databaselike data in a way in which subblocks are linked to or "filed underneath" their parent blocks. In reality it would seem that they're still using a broadly defined index card type system as used in the late 1800s/early 1900s to implement a set up that otherwise would be a traditional database in the Microsoft Excel or MySQL sort of fashion, the major difference being that the user interface is cognitively easier to understand for most people.
These allow people to take a form of structured textual notes to which might be attached other smaller data or meta data chunks that can be easily searched, sorted, and filtered to allow for quicker or easier use.
Ostensibly from a mathematical (or set theoretic and even topological) point of view there should be a variety of onetoone and onto relationships (some might even extend these to "links") between these sorts of notes and database representations such that one should be able to implement their note taking system in Excel or MySQL and do all of these sorts of things.
Cascading Idea Sheets or Cascading Idea Relationships
One might analogize these sorts of note taking interfaces to Cascading Style Sheets (CSS). While there is the perennial question about whether or not CSS is a programming language, if we presume that it is (and it is), then we can apply the same sorts of class, id, and inheritance structures to our notes and their meta data. Thus one could have an incredibly atomic word, phrase, or even number(s) which inherits a set of semantic relationships to those ideas which it sits below. These links and relationships then more clearly define and contextualize them with respect to other similar ideas that may be situated outside of or adjacent to them. Once one has done this then there is a variety of Boolean operations which might be applied to various similar sets and classes of ideas.
If one wanted to go an additional level of abstraction further, then one could apply the ideas of category theory to one's notes to generate new ideas and structures. This may allow using abstractions in one field of academic research to others much further afield.
The user interface then becomes the key differentiator when bringing these ideas to the masses. Developers and designers should be endeavoring to allow the power of complex searches, sorts, and filtering while minimizing the sorts of advanced search queries that an average person would be expected to execute for themselves while also allowing some reasonable flexibility in the sorts of ways that users might (most easily for them) add data and meta data to their ideas.
Jupyter programmable notebooks are of this sort, but do they have the same sort of hierarchical "card" type (or atomic note type) implementation?
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 category theory
 Roam Research
 Beatrice Webb
 programmable notes
 CSS
 cascading idea sheets
 types of notes
 super tags
 user interface
 card index as database
 building blocks
 Boolean algebra
 Jupyter
 scientific note taking
 Tana
 integrated development environment
 idea links
 Maggie Delano
 integrated thinking environments
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 Aug 2022

www.sciencedirect.com www.sciencedirect.com

Corner, A., Hahn, U., & Oaksford, M. (2011). The psychological mechanism of the slippery slope argument. Journal of Memory and Language, 64(2), 133–152. https://doi.org/10.1016/j.jml.2010.10.002


www.venturekit.com www.venturekit.com

a petal competition slide, pioneered by Steve Blank. It’s intended for companies creating a new market.


www.dreamit.com www.dreamit.com

when an investor sees a Magic Quadrant, they'll think:Can your company actually only differentiate its product on two axes? Does your product beat the competition on just two key benefits or strategies?If that's the case, and your product isn't much different from your competitors, why will people buy it or use it?

 Jan 2022

www.youtube.com www.youtube.com

https://www.youtube.com/watch?v=z3Tvjf0buc8
graph thinking
 intuitive
 speed, agility
 adaptability
; graph thinking : focuses on relationships to turn data into information and uses patterns to find meaning
property graph data model
 relationships (connectors with verbs which can have properties)
 nodes (have names and can have properties)
Examples:
 Purchase recommendations for products in real time
 Fraud detection
Use for dependency analysis

 Sep 2021

psyarxiv.com psyarxiv.com

Motz, B., Fyfe, E., & Guba, T. P. (2021). Learning to Call Bullsh*t via Induction: Categorization Training Improves Critical Thinking Performance. PsyArXiv. https://doi.org/10.31234/osf.io/65qfj


www.mdpi.com www.mdpi.com

The CommunitySensor community network ontology can be positioned somewhere in the middle of this spectrum: community network representatives are totally free to come up with their own terms for element and connection types in their own ontologies. However, these terms are organized in a deep structure with communityspecific element and connection types being classified by higherorder element and connection type (sub)categories described in the CommunitySensor community network conceptual model.
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 Aug 2021

www.agilemodeling.com www.agilemodeling.com

Figure 1. The relationship between models, documents, source code, and documentation.

 Jul 2021

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 Mar 2021

en.wikipedia.org en.wikipedia.org

en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org

In fact categories can themselves be viewed as type theories of a certain kind


en.wikipedia.org en.wikipedia.org

Mathematically, a hierarchical taxonomy is a tree structure of classifications for a given set of objects.

 Feb 2021

en.wikipedia.org en.wikipedia.org

Though rarer in computer science, one can use category theory directly, which defines a monad as a functor with two additional natural transformations. So to begin, a structure requires a higherorder function (or "functional") named map to qualify as a functor:
rare in computer science using category theory directly in computer science What other areas of math can be used / are rare to use directly in computer science?


dryrb.org dryrb.org

It's hard to say why people think so because you certainly don't need to know category theory for using them, just like you don't need it for, say, using functions.


docdrop.org docdrop.org

Certificado
category

 Oct 2020

www.quantamagazine.org www.quantamagazine.org

In some sense, by studying one model deeply enough, we can study them all.
This may be where math like category theory is particularly powerful as a map between these different areas which are really the same (isomorphic).

 Aug 2020

www.robinsloan.com www.robinsloan.com

First, what would it look like for a social media platform to reestablish perspective?
This was the exact design question I asked recently!

Browsing Twitter the other day, I once again found myself sucked into a faroff event that truly does not matter, and it occurred to me that social media is an orthographic camera.
How does this relate to Nicholas Carr's article and ideas about category errors in From context collapse to content collapse?


en.wikipedia.org en.wikipedia.org

A category mistake, or category error, or categorical mistake, or mistake of category, is a semantic or ontological error in which things belonging to a particular category are presented as if they belong to a different category,[1] or, alternatively, a property is ascribed to a thing that could not possibly have that property.


www.roughtype.com www.roughtype.com

A fascinating viewpoint on social media, journalism, and information. There are some great implied questions for web designers hiding in here.

In discussing the appeal of the News Feed in that same interview with Kirkpatrick, Zuckerberg observed, “A squirrel dying in front of your house may be more relevant to your interests right now than people dying in Africa.” The statement is grotesque not because it’s false — it’s completely true — but because it’s a category error. It yokes together in an obscene comparison two events of radically different scale and import. And yet, in his tonedeaf way, Zuckerberg managed to express the reality of content collapse. When it comes to information, social media renders category errors obsolete.
How can we minimize this sort of bias? How can we help to increase the importance of truly important things?


covid19.iza.org covid19.iza.org

Which Jobs Are Done from Home? Evidence from the American Time Use Survey. COVID19 and the Labor Market. (n.d.). IZA – Institute of Labor Economics. Retrieved August 8, 2020, from https://covid19.iza.org/publications/dp13138/

 Jun 2020

psyarxiv.com psyarxiv.com

Chen, C., Messinger, D., Duan, Y., Ince, R. A., Garrod, O. G. B., schyns, P., & Jack, R. (2020). Dynamic facial expressions of emotion decouple emotion category and intensity information over time [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/4gpev

 May 2020

misinforeview.hks.harvard.edu misinforeview.hks.harvard.edu

Makhortykh, M., Urman, A., & Ulloa, R. (2020). How search engines disseminate information about COVID19 and why they should do better. Harvard Kennedy School Misinformation Review, 1(COVID19 and Misinformation). https://doi.org/10.37016/mr2020017

 Apr 2020

en.wikipedia.org en.wikipedia.org

In category theory
'
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ruby.libhunt.com ruby.libhunt.com

www.nature.com www.nature.com

Morey, R.A., Haswell, C.C., Stjepanović, D. et al. Neural correlates of conceptuallevel fear generalization in posttraumatic stress disorder. Neuropsychopharmacol. (2020). https://doi.org/10.1038/s4138602006618

 Jan 2020

en.wikipedia.org en.wikipedia.org

en.wikipedia.org en.wikipedia.org

en.wikipedia.org en.wikipedia.org
 Oct 2019


categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations. In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.
monoidal (category > computer)
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 Sep 2019


from falsehood you can derive everything ** false \leq truerestrict: don't talk about elements > you have to talk about arrows (relations) .... interview the friends *product types: [pairs, tuples, records,...]

 Aug 2019

bartoszmilewski.com bartoszmilewski.com

But there is an alternative. It’s called denotational semantics and it’s based on math. In denotational semantics every programing construct is given its mathematical interpretation. With that, if you want to prove a property of a program, you just prove a mathematical theorem


www.math3ma.com www.math3ma.com

hierarchy of questions: "What about the relationships between the relationships between the relationships between the...?" This leads to infinity categories. [And a possible brain freeze.] For more, see here.) As pieinthesky as this may sound, these ideascategories, functors, and natural transformationslead to a treasure trove of theory that shows up almost everywhere.
Turtles all the way up

 Aug 2018

emlis.pair.com emlis.pair.com

Time as order considers temporal data as data that can be described by a time attribute, which can improve navigation or organization. Time as change considers temporal data as data that evolve over time.
The contrast of time as representing either order or change could be a very helpful way to categorize messy, dynamic humanitarian crisis data.
We need ways to capture data as an event timeline (X happened, then Y which demands response Z) as well acknowledge that X and Y may be in flux in an evolving crisis zone.

 Jul 2018

arxiv.org arxiv.org

“pulls it back”
minor quibble, maybe this should be surrounded by parantheses
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 Jun 2018

arxiv.org arxiv.org

Exercise1.75.Doesbù3chave a right adjointR:N!N? If not, why? If so, does itsright adjoint have a right adjoint?

Remark1.73.IfPandQare total orders andf:P!Qand1:Q!Pare drawn witharrows bending as in Exercise 1.72, we believe thatfis left adjoint to1iff the arrows donot cross. But we have not proved this, mainly because it is difficult to state precisely,and the total order case is not particularly general

The preservation of meets and joins, and hence whether a monotone map sustainsgenerative effects, is tightly related to the concept of a Galois connection, or moregenerally an adjunction.

Galois connections between posets were first considered by Évariste Galois—whodidn’t call them by that name—in the context of a connection he found between “fieldextensions” and “automorphism groups”. We will not discuss this further,

In his work on generative effects, Adam restricts his attention to maps that preservemeets, even while they do not preserve joins. The preservation of meets implies that themapbehaves well when restricting to a subsystem, even if it can throw up surpriseswhen joining systems

n [Ada17], Adam thinks of monotone maps as observations. A monotone map:P!Qis a phenomenon ofPas observed byQ. He defines generative effects of such a mapto be its failure to preserve joins (or more generally, for categories, its failure topreserve colimits)

Example1.61.Consider the twoelement setPfp;q;rgwith the discrete ordering.The setAfp;qgdoes not have a join inPbecause ifxwas a join, we would needpxandqx, and there is no such elementx.Example1.62.In any posetP, we havep_pp^pp.Example1.63.In a power set, the meet of a collection of subsets is their intersection,while the join is their union. This justifies the terminology.Example1.64.In a total order, the meet of a set is its infimum, while the join of a set isits supremum.Exercise1.65.Recall the division ordering onNfrom Example 1.29: we say thatnmifndivides perfectly intom. What is the meet of two numbers in this poset? Whatabout the join?
These are all great examples. I htink 1.65 is gcd and lcm.

These notions will have correlates in category theory, called limits and colimits,which we will discuss in the Chapter 3. For now, we want to make the definition ofgreatest lower bounds and least upper bounds, called meets and joins, precise.

Ifxyandyx, we writexyand sayxandyareequivalent. We call a set with a preorder aposet.

Example1.49.Recall from Example 1.36 that given a setXwe defineEXto be theset of partitions onX, and that a partition may be defined using a surjective functions:XPfor some setP.Any surjective functionf:X!Yinduces a monotone mapf:EY! EX, going“backwards”. It is defined by sending a partitions:YPto the compositef:s:XP

Example1.42 (Opposite poset).Given a posetπP;∫, we may define the opposite posetπP;op∫to have the same set of elements, but withpopqif and only ifqp.

Example1.40 (Product poset).Given posetsπP;∫andπQ;∫, we may define a posetstructure on the product setPQby settingπp;q∫ πp0;q0∫if and only ifpp0andqq0. We call this theproduct poset. This is a basic example of a more generalconstruction known as the product of categories

Contrary to the definition we’ve chosen, the term poset frequently is used to meanpartiallyordered set, rather than preordered set. In category theory terminology, therequirement thatxyimpliesxyis known asskeletality. We thus call partiallyordered setsskeletal posets

 Sep 2016

bartoszmilewski.com bartoszmilewski.com

ategory consists of objects and arrows that go between them
Based on the arrow being a morphism/function (see below), it seems that the domain and the codomain should be able to be disjoint.
Also, here A>B and A>C; the dual map means that an arrow is not a function.

 Dec 2015

math.mit.edu math.mit.eduCT4S.pdf16

Lemma 2.5.1.14
Invoke universal property of products

Since ducks can both swim and fly, each duck is found twice inC, once labeled as aflyer and once labeled as a swimmer. The typesAandBare kept disjoint inC, whichjustifies the name “disjoint union.”
The disjoint union reminds me of algebraic datatypes in functional programming languages, whereas a settheoretic union is more like a union in CS: the union has no label associated with it, so additional computation (or errors) may arise due to a lack of ready information about elements in the union.

facts, which are simply “path equivalences” in an olog. It isthe notion of path equivalences that make category theory so powerful.Apathin an olog is a headtotail sequence of arrows

Consider the aspectpan objectqhas››››—pa weightq. At some point in history, thiswould have been considered a valid function. Now we know that the same objectwould have a different weight on the moon than it has on earth. Thus as worldviews change, we often need to add more information to our olog. Even the validityofpan object on earthqhas››››—pa weightqis questionable. However to build a modelwe need to choose a level of granularity and try to stay within it, or the whole modelevaporates into the nothingness of truth!

An aspect of a thingxis a way of viewing it, a particular way in whichxcan be regardedor measured. For example, a woman can be regarded as a person; hence “being a person”is an aspect of a woman. A molecule has a molecular mass (say in daltons), so “havinga molecular mass” is an aspect of a molecule. In other words, byaspectwe simply meana function. The domainAof the functionf:A—Bis the thing we are measuring, andthe codomain is the set of possible “answers” or results of the measurement.
Naïvely (since my understanding of type theory is naïve), this seems to mesh with the concepts of inheritance for the "is" relationships, and also with typetheory more generally for "has" relationships, since I believe we can view any object or "compound type", as defined here, as being a subtype of another type 'o' if one of its elements is of type 'o'. Though we have to be careful for functional mapping when thinking of aspects: we can't just say Int is an aspect of Pair(Int, Int), since this is ambiguous (there are two ints)  we must denote which Int we mean.

We represent eachtype as a box containing asingular indefinite noun phrase.

Data gathering is ubiquitous in science. Giant databases are currently being minedfor unknown patterns, but in fact there are many (many) known patterns that simplyhave not been catalogued. Consider the wellknown case of medical records. A patient’smedical history is often known by various individual doctoroffices but quite inadequatelyshared between them. Sharing medical records often means faxing a handwritten noteor a filledin housecreated form between offices.

As mentioned above category theory has branched out into certain areas of scienceas well. Baez and Dolan have shown its value in making sense of quantum physics, itis well established in computer science, and it has found proponents in several otherfields as well. But to my mind, we are the very beginning of its venture into scientificmethodology. Category theory was invented as a bridge and it will continue to serve inthat role.

All this time, however, category theory was consistently seen by much of the mathematical community as ridiculously abstract. But in the 21st century it has finally cometo find healthy respect within the larger community of pure mathematics. It is the language of choice for graduatelevel algebra and topology courses, and in my opinion willcontinue to establish itself as the basic framework in which mathematics is done

In 1980 Joachim Lambek showed that the types and programs used in computerscience form a specific kind of category. This provided a new semantics for talking aboutprograms, allowing people to investigate how programs combine and compose to createother programs, without caring about the specifics of implementation. Eugenio Moggibrought the category theoretic notion of monads into computer science to encapsulateideas that up to that point were considered outside the realm of such theory.

Bill Lawvere saw category theory as a new foundation for all mathematical thought.Mathematicians had been searching for foundations in the 19th century and were reasonably satisfied with set theory asthe foundation. But Lawvere showed that the categoryof sets is simply a category with certain nice properties, not necessarily the center ofthe mathematical universe. He explained how whole algebraic theories can be viewedas examples of a single system. He and others went on to show that higher order logicwas beautifully captured in the setting of category theory (more specifically toposes).It is here also that Grothendieck and his school worked out major results in algebraicgeometry.
I haven't studied toposes, but I can at least see how introductory algebraic geometry, i.e. the study of Groebner bases, relates to propositional logic.

The paradigm shift brought on by Einstein’s theory of relativity brought on the realization that there is no single perspective from which to view the world. There is nobackground framework that we need to find; there are infinitely many different frameworks and perspectives, and the real power lies in being able to translate between them.It is in this historical context that category theory got its start.

These theorems have not made theirway out into the world of science, but they are directly applicable there. Hierarchies arepartial orders, symmetries are group elements, data models are categories, agent actionsare monoid actions, localtoglobal principles are sheaves, selfsimilarity is modeled byoperads, context can be modeled by monads.

No one would dispute that vector spaces are ubiquitous.But so are hierarchies, symmetries, actions of agents on objects, data models, globalbehavior emerging as the aggregate of local behavior, selfsimilarity, and the effect ofmethodological context.

I will use a mathematical tool calledologs, or ontology logs, to givesome structure to the kinds of ideas that are often communicated in pictures like theone on the cover. Each olog inherently offers a framework in which to record data aboutthe subject. More precisely it encompasses adatabase schema, which means a system ofinterconnected tables that are initially empty but into which data can be entered.

Agreementis the good stuff in science; it’s the high fives.But it is easy to think we’re in agreement, when really we’re not. Modeling ourthoughts on heuristics and pictures may be convenient for quick travel down the road,but we’re liable to miss our turnoff at the first mile. The danger is in mistaking ourconvenient conceptualizations for what’s actually there. It is imperative that we havethe ability at any time to ground out in reality.
