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have created geometric infinity-stack
gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in ) and indicated the possibility of another definition, along the lines that we are discussing on the Café
added to quantum anomaly
an uncommented link to Liouville cocycle
a paragraph with the basic idea of fermioninc anomalies
the missing reference to Witten’s old article on spin structures and fermioninc anomalies.
The entry is still way, way, stubby. But now a little bit less than a minute ago ;-
the table didn’t have the basic examples, such as Gelfand duality and Milnor’s exercise. Added now.
following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.
The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.
I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.
(Hi, I’m new)
I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the empty (co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.
I also noticed that the page about the empty space referred to the naive definition of connectedness as being
“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”
but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).
adding references
Ming Ng, Steve Vickers, Point-free Construction of Real Exponentiation, Logical Methods in Computer Science, Volume 18, Issue 3 (August 2, 2022), (doi:10.46298/lmcs-18(3:15)2022, arXiv:2104.00162)
Steve Vickers, The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms, (arXiv:2312.05228)
Anonymouse
brief category:people-entry for hyperlinking references at equivariant principal bundle
category: people page for the reference
Anonymouse
category: people page for the reference
Anonymouse
I added to excluded middle a discussion of the constructive proof of double-negated LEM and how it is a sort of “continuation-passing” transform.
I’ve been inactive here for some months now; I hope this will significantly change soon.
I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:
Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?
Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?
In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products , and form a new 2-category with 2-products whose 0-cells are pseudomonoids in , whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate , starting with . Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace with (where all the units are forced to coincide up to isomorphism)?
Apologies if these are naive questions; I am not very familiar with the literature.
a bare minimum, for the moment just so as to satisfy links from graded modality
I think the line between the two types of Kan extension (weak versus pointwise) is drawn at the wrong place. Am I missing something?
copied over the homotopy-theoretic references from modal type theory to here.
http://ncatlab.org/nlab/show/Isbell+duality
Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality
However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf
seems to use another definition. Could someone please clarify?
added to G2 the definition of as the subgroup of that preserves the associative 3-form.
collected some references on the interpretation of the !-modality as the Fock space construction at !-modality.
Cross-linked briefly with he stub entries_Fock space_ and second quantization.
Added to noetherian ring a homological chacaterization: a ring is Noetherian iff arbitrary direct sums of injective modules are injective.