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Apr
1
comment defining a bicategory of real-valued matrices
Thank you, this is helpful. I am still interested in the original example (in particular, real-valued matrices with ordinary matrix multiplication) and whether it can be given something like the structure of a proarrow equipment, but it's helpful to have spelled out how this example differs from $\mathbf{Rel}$.
Mar
31
comment defining a bicategory of real-valued matrices
$\mathbf{FinMat}$ is a monoidal category, and so can be re-interpreted as a 2-category in the way you describe, but that just shifts my question one dimension up. Under that interpretation, each finite function $f : X \to Y$ determines a pair of linear transformations $k^{|X|} \to k^{|Y|}$ and $k^{|Y|} \to k^{|X|}$, and the question is whether/in what sense these can be seen as "adjoint"?
Mar
31
asked defining a bicategory of real-valued matrices
Feb
14
comment Does there exist a terminal surjective discrete fibration out of $C$?
Suppose that $C$ is a monoid, viewed as a one-object category. A discrete opfibration $F : C \to D$ over another one-object category $D$ corresponds to an injective homomorphism, and so there is an initial such one corresponding to the identity functor $C \to C$. What do you have in mind for the terminal discrete opfibration from $C$?
Jan
19
awarded  Necromancer
Dec
7
comment efficient arithmetic with (short) Conway games?
yes, I see now that Conway mentions this in "More Infinite Games" (library.msri.org/books/Book42/files/conway.pdf), though I'm curious as to why there is a valid definition of multiplication for Nimbers/impartial games.
Dec
6
comment efficient arithmetic with (short) Conway games?
I don't have a copy of the book yet, but I see that CGSuite fails evaluating multiplications like "{0|{0|0}} * {0|0}" ('No method "op *" for class CanonicalShortGame.'). Is the general case of arithmetic with short games still open (or known to be hard)?
Dec
5
comment efficient arithmetic with (short) Conway games?
This is great, thanks! I downloaded CGSuite, and will have a look at the book.
Dec
5
accepted efficient arithmetic with (short) Conway games?
Dec
5
asked efficient arithmetic with (short) Conway games?
Nov
16
awarded  Citizen Patrol
Nov
13
awarded  Revival
Nov
13
awarded  Yearling
Nov
13
answered Relation between Metalanguage and Object Language
Nov
8
comment Are paths in HoTT perhaps just “cost-free” paths?
@DavidSpivak Okay. As I wrote in my answer, I think the basic idea of "mutations as paths with costs" makes sense, and corresponds well with the basic judgments of Hoare logic (where the "cost" of a mutation is the command by which it is realized). However, in that interpretation, cost-free path = vertical morphism, and so I don't quite see where you are going in defining the space $X_0$. Would you still apply this construction if weights were generalized to be maps of an arbitrary category, rather than elements of a monoid?
Nov
6
comment Are paths in HoTT perhaps just “cost-free” paths?
@DavidSpivak can you explain what you meant by "identities" in Q1?
Nov
5
revised Are paths in HoTT perhaps just “cost-free” paths?
added some explanation
Nov
5
comment Are paths in HoTT perhaps just “cost-free” paths?
@NoahS 𝔹 is the category of state types and state transformers. Traditionally in Hoare logic there is only state type, hence 𝔹 is a one-object category, i.e., a monoid (this restriction is not really necessary though). 𝔼 is the category of "predicates over states and proofs of Hoare triples". For example, if we identify predicates with subsets, then an arrow of 𝔼 from $P \subseteq S$ to $Q \subseteq S$ corresponds to a state transformer $c : S \to S$ such that $c(P) \subseteq Q$. Finally, $p : \mathbb{E} \to \mathbb{B}$ is just the forgetful functor.
Nov
4
revised Are paths in HoTT perhaps just “cost-free” paths?
notation/wording
Nov
4
answered Are paths in HoTT perhaps just “cost-free” paths?