Categorical foundations: special topics in order, topology, algebra, and Sheaf theory
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Applications of Categorical Algebra. Topics in Algebra. Topics in algebra. Topics in Universal Algebra.
Topics in Low-Dimensional Topology. Categorical Logic and Type Theory. Special Topics in Calamity Physics. Recommend Documents. Artin, H. Bass, J. First, you need to understand the category of sets s being a well-pointed topos, internal to the syntactic bi -category of predicates and functional predicates classes. For this you want to look at. Second, algebra is really about monads, in the sense that any category of algebraic objects is a category of algebras for a monad.
For this you should read. Next, you will need some knowledge of enriched category theory in monoidal closed categories since, since most of the monads of basic algebra comes from monoid objects in a monoidal closed category e. The relevant material for understanding the constructions of these categories are the first few chapters of. It is here, in considering the enriched category theory perspective, where having a good understanding of the category of sets as a well-pointed topos is crucial. Without well-pointedness, you cannot conclude much about the categories of functors that you are building, and which the various categories of algebraic objects ultimately are.
Finally, there are the papers "Monads on Symmetric Monoidal Categories" and "Closed Categories Generated by Commutative Monads" by Anders Kock that address the fact that algebras for commutative monads inherit a monoidal closed structure when the commutative monad is in a monoidal closed category. This is where tensor products, for example, really come from. Regarding topology, the Categorical Foundations book also has Chapter III: A Functional Approach to General Topology, which is quite enlightening, but you should probably only read it concurrently with an actual topology book, like Munkres.
Ronald Brown's text Topology and Groupoids is probably what you want from a topology text.
Special order items
He gives an introduction to general topology and the fundamental groupoid using the language of category theory throughout. It's an excellent textbook. I would also second other posters' recommendations of Aluffi's algebra textbook; it's very well-written and beginner-friendly. For homological algebra, I would recommend Weibel's text. If you can understand German and according to your original post I assume so , for an introduction to topology with a touch of category theory I'd avice you to have a look at "Grundkurs Topologie" by Gerd Laures and Markus Szymik.
It might just be what you are looking for. Although I doubt you will see Yoneda "in action". I certainly asked myself about that some time ago I'm still an undergraduate student , and I found Aluffi's: Algebra Chapter 0 to be the most exciting, interesting and categorical introduction to abstract algebra which I know of. I must say that I even hated everything related to algebra during my first undergraduate months; then I took linear algebra and it was not that bad, but it was during an advanced linear algebra course where I suddenly learned several cool things about rings, modules, diagram chasing and canonical forms from an advanced point of view such as how the Krull-Schmidt theorem is involved within the uniqueness of such descompositions.
It was very hard for me as a second year student to grasp the course without being very exhausted, but it was absolutely awesome, so I drastically began to change my feelings for abstract algebra forever. At this point I discovered Aluffi's book and it became a timeless classic for me since the first moment.
I think every serious student should take a look at this book, no matter if they like algebra or not. I wish I've came across this book earlier, and by the way, don't be intimitated by the fact that it is considered to be a graduate-level textbook, it might serve as an advanced and complete reference for undergraduate courses and students as well. As for the topology book, I want to recommend a book which was written by my topology professor himself. As it was pointed out earlier, Aguilar, Gitler and Prieto's Algebraic Topology from an Homotopical Point of View is a very nice algebraic topology book which uses several gadgets from category theory whenever possible.
It might be a very good book for the algebraically-minded student who is taking a first course on topology or anyone with interest on learning algebraic topology as it covers the basic point-set topology material as well. Fortunately, there is an english translation of the book which is split into two parts: Elements of point-set topology and Elements of homotopy theory. As far as I can tell I agree with lentic catachresis when he says that Aluffi's book is a very good introduction to algebra in a categorical setting, although I've found that Lang's text book is a good reference too, especially for more advanced topics.
Any book in homological-algebra makes intense use of category theory, which isn't that a surprise considering that category theory was born for solving problems in these fields. From the topological point of view, I've studied from Manetti Topology. In my opinion it is a really good introductory book on general topology with a categorical perspective: many concepts are presented and emphatized from arrow-point-of-view. It also has the same limitation of Aluffi's book: it doesn't make use of anything more advanced of limits and universal properties.
If you want to see more advanced application of category theory Spanier's Algebraic Topology makes use of stuff like Yoneda lemma, although some time these application are not made explicit.
Another very good reference about application of category theory to algebraic topology is Aguilar, Gitler and Prieto's Algebraic Topology from an Homotopical Point of View : in this book you can really see a lot of applications of category theory to topology as an example, if I remember correctly, there is a proof of the fact that the category of compactly generated spaces is cartesian closed via the adjoint functor theorem.
You should give a look to Categorical Foundations , by Pedicchio and Tholen. I wrote a book "Introductory Algebra, Topology, and Category Theory" in , exactly addressing these issues. It's currently available for free at www. Chapters 2 and 3 cover universal algebra and order theory. Another challenge, then is to model various aspects of transitive inference, including pseudo-transitive inference Goodwin and Johnson-Laird, , where the elements of the premises are locally, but not globally ordered.
A sheaf theory approach may also have something to say about the development of transitive inference and other reasoning tasks in terms of the development of the underlying topological space. Young children below about 5 years of age repeatedly have been shown to lack a capacity for transitive inference and a range of other reasoning tasks Halford, ; Andrews and Halford, , Some have argued that such capacities turn on the development of relational information processes Halford et al.
The category theory perspective attributed the difference to a capacity for products, including constrained products pullbacks. We have already seen how these constructions are related to presheaves and sheaves, and the underlying topology. The sheaf theory approach presented here provides another related perspective on the development and evolution of intelligence, i.
In particular, we noted that every set has two extreme topologies: indiscrete and discrete. For the collection of topological spaces on a given set, the indiscrete and discrete spaces are, respectively, the coarsest and finest topologies that can be given for that set, which are themselves instances of particular universal constructions.
For example, young children represent changes in shape differently than adults Abecassis et al. In our sheaf theory view, intersection discovery connotes development of a topological space. DORA uses the role-filler binding method of the LISA model Hummel and Holyoak, to induce relational representations via the interaction between proposition units representing relations, role-filler units representing the binding of values to relational roles, and feature units representing features of the related fillers values —role-filler units that coactivate the same feature units tend to be bound together by units representing a common relation.
The dynamics of the DORA model are more complex than projections. So, the extent of a formal connection is not yet known. Developing a neural model for the theory would provide a basis for cost in terms of the neural resources needed to realize a universal construction. Whether similar considerations apply to the development of conjunction search is a topic for future work.
A capacity to represent conjunctions is just one aspect of visual attention, and there are multiple possible reasons for a change in search efficiency with age see Merrill and Lookadoo, , for a discussion. Thus, changing C which means changing f and g can change the number of elements constructed by the pullback, hence the number of elements selected for search, and thereby search efficiency. Other potential applications are probability judgements that violate classical probability laws, e. In this situation, people judge the conjunction of two events A and B as more likely than either event A or event B : e.
Quantum probability theory was introduced to explain such fallacies see Busemeyer and Bruza, , for an overview of theory and example applications. An important feature of this theory is contextuality where the act of measuring affects the outcome. The conditions for having quantum-like contextuality effects are closely related to the conditions for being a presheaf, but not a sheaf Abramsky and Brandenburger, In these situations, the points of the topological space are measures and the values data attached to the space our outcomes, or outcome probabilities.
Presheaves involve three kinds of morphisms, in addition to inclusions and restrictions: 1 morphisms from the topological space to the data, i. We have primarily concerned ourselves with the second kind, in the form of sheaving, with regard to the generalization aspect of learning. However, for a more complete picture, we also need to consider how the first and third kinds of morphism pertain to other aspects of learning.
The first kind of morphism is important with regard to training and the partial state of knowledge acquisition. In particular, one difficulty with a category theory approach to cognition is how to model partial knowledge Navarrete and Dartnell, With sheaf theory, partial knowledge can be related to the data attached to open sets and their restrictions. This situation reflects a temporary state of having partial no knowledge about, or representation of color-shape binding.
In the context of learning, partial acquisition of knowledge can be modeled as a subset of the sections rows attached to an open set. How data get attached to open sets as a result of learning is a topic of future work. The third kind of morphism is important in regard to explaining the transition from non-systematicity to systematicity. As mentioned in section 2. Throughout this paper, we have focussed only on interpreting the gluing condition for a sheaf as a formal basis for the ability to go beyond the data. However, a presheaf must also satisfy the locality condition to be a sheaf.
The locality condition says that gluing must be unique. A detailed exposition will take us too far afield, however, this situation is like treating different instances of an object as the same object up to some equivalence relation. This situation typically does not arise in a relational database, because the relational database schema is essentially treated as a discrete topological space, in which case all rows must be unique, i. In a cognitive context, the locality condition may also have interpretations in terms of treating two distinct entities as the same thing: generalization on the basis of object class, rather than object instance.
Some authors have argued that the architecture of cognition i. Be that as it may, viewing cognitive architecture as a hodgepodge of subsystems begs the question of why the system does actually work coherently, for the most part. The sheaf theory view presented here says that patching is a universal construction: an optimal solution to reconciling differences between subsystems put together as a kludge. A sheaf can be likened to a kind of analogy in that the relations inclusions in the source domain topological space are mapped to relations restrictions in the target domain data attached to the space , cf.
Category theory has been used as a basis for children's difficulty with understanding and exploiting the common relations in a reasoning problem Halford and Wilson, , and as an approach to analogy Navarrete and Dartnell, However, a sheaf is a contravariant functor: the directions of arrows in the source are reversed in the target, which may strike some people as puzzling, given that analogy is typically conceptualized as a covariant mapping: the directions of arrows in the source and target are the same. One can conceptualize the role of contravariance in sheaf theory as persistence. Topological spaces can be built up by taking intersections and unions of the open sets in a basis set.
Inclusions order open sets by size from small to large. A global property can be regarded as a property that persists over all the open sets—a property that is systematic as opposed to idiosyncratic specific to just some open sets—as we zoom in on smaller regions of space. An apparently straightforward approach would be to assign a cost to the alternative routes see Phillips and Wilson, b , for a discussion.
Independent motivation may come in the form of empirical measures of the cost of each supposed alternative. For such purposes, a split-screen paradigm was developed to examine cost in the context of feature vs. In this paradigm, participants could search for the target object in either the left or right visual field, which corresponded to feature or conjunction search. Search time when only one field was presented provided independent measures of the baseline costs of feature and conjunction search, which were then used to assess whether participants chose the alternative of least cost when both alternatives where presented at the same time.
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Analysis indicated that the choice of search field depended not only on the relative costs of the alternatives, but also on the cost of that assessment Phillips et al. Going beyond the data is a ubiquitous cognitive capacity in need of a theoretical explanation to motivate modeling as more than just an exercise in data fitting. The theoretical picture painted here is a view beyond local perception of the world. This sheaf theory approach formalizes our propensity to connect the dots. After all, that's what people do. The author confirms being the sole contributor of this work and approved it for publication.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
National Center for Biotechnology Information , U. Journal List Front Psychol v. Front Psychol. Published online Oct 9. Author information Article notes Copyright and License information Disclaimer. This article was submitted to Theoretical and Philosophical Psychology, a section of the journal Frontiers in Psychology. Received Jun 7; Accepted Sep The use, distribution or reproduction in other forums is permitted, provided the original author s and the copyright owner s are credited and that the original publication in this journal is cited, in accordance with accepted academic practice.
No use, distribution or reproduction is permitted which does not comply with these terms. Keywords: learning, generalization, sheaf theory, sheaf, sheaving, category theory, universal. Preview: generalization as patching Sheaving A capacity to generalize beyond the given instances connotes a property that is re constructed from local information.
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Sheaves and sheaving As previewed in the Introduction, sheaf theory concerns the passage from local to global properties, which we interpret as generalization in the context of cognition. Open in a separate window. A topological view of space The street map example may leave one wondering about the need to work with a topological space. A relational view of data The foregoing conception of representational space lays the groundwork, as it were, for a parsimonious treatment of representation as data attached to a topological space.
Figure 1. Gluing as a constrained product The essential difference between gluing as a product and gluing as a constrained product is that the intersection of the underlying open sets, to which the data are attached, is not the empty set. Figure 2. An example of sheaving as a constrained product empty box indicates empty set. Going beyond the data: sheaving in cognition The sheaf theory constructions just presented are applied to cognitive domains. Cue-target learning: product In this section, we show why generalization is afforded by sheaving for a task requiring participants to learn a set of cue-target mappings that is the product of two sets of cue-target mappings Phillips et al.
Figure 3. Visual search: constrained product In a visual search task, participants are required to locate an object, designated as the target of search, in a display also containing nontargets. Author contributions The author confirms being the sole contributor of this work and approved it for publication. Conflict of interest statement The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Footnotes Funding. References Abecassis M. What's in a shape: children represent shape variability differently than adults when naming objects.
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Categorical foundations: Special topics in order, topology, algebra, and sheaf theory
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