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# Bruno Gavranović

Computer science Master student at the University of Zagreb, Croatia. I'm interested in understanding how intelligence works in the most universal way possible - which seems to be the language of category theory. I have plenty of experience with deep neural networks (see my github ) and now I'm attending the ACT course and getting solid foundations in CT. I'm happy to talk about anything related to CT or machine learning - feel free to drop me a message!

Bruno Gavranović
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• Do you mean the "Formally, it is a quotient object in the category of (small) categories, analogous to a quotient group or quotient space, but in the categorical setting." sentence? Doesn't a path equivalence relation on Free(G) exactly the congruen…
• There is also a phrase "finitely-presented category" mentioned in Seven sketches, which seems like a very similar thing. I'm wondering if there are any subtle differences
• I'd also add three recent very interesting papers: On Characterizing the Capacity of Neural Networks using Algebraic Topology - https://arxiv.org/abs/1802.04443 Backprop as Functor - https://arxiv.org/abs/1711.10455 The simple essence of automati…
• Thanks Scott for the extensive answer! I do realize I was very loose with the terminology, changing between monoidal and cartesian product and referring to morphisms as functions. I hope I did manage to get the point across, as these errors seem to …
• Thank you so much for the lectures, John! I've been following the course since day 1 and it has been a tremendous help. I'm just studying a master in computer science so I've been lacking a lot of the mathematical maturity people seem to have here …
• Form the Cost-product $$\mathbb{R} \times \mathbb{R}$$: $$Ob(\mathcal{X}) = \mathbb{R} \times \mathbb{R}$$ $$(\mathcal{X} \times \mathcal{Y})((x_1, y_1), (x_2, y_2)) = \mathcal{X}(x_1, x_2) \otimes \mathcal{Y}(y_1, y_2) = |x_1 - x_2| + |y_1 -… • @Valter, that's exactly where I saw the hypercube completion quote! What he drew in that post is basically a 3d commutative diagram - it just goes a long way to show that people seem to use categorical way of thinking without realizing it. Maybe eve… • I'm having trouble understanding what a dagger Cost-category is. From the definition above I understand what a dagger structure on a category is: it's a special type of a functor. But I'm not sure how to interpret it as a part of my metric space \(… • This is the second quasi-inverse function from the exercise 2.41: [ g(x)= \begin{cases} \mathtt{true}, & \text{if x < \infty}.\\ \mathtt{false}, & \text{if  x = \infty}. \end{cases} ] Both of them map \( 0$$ to $$\mathtt{true}$$ a…
• Using the function [ f(x)= \begin{cases} \mathtt{true}, & \text{if $x = 0$}.\\ \mathtt{false}, & \text{if $x > 0$}. \end{cases} ] we can define $$\mathcal{X}(x, y)$$ for our any $$x, y \in \mathcal{X}$$ in our "regions of the wor…
• @Fredrick, multiplying the matrix by itself (using the Eq. 2.97 that tells us about matrix multiplication in a quantale) yields a matrix that tells us about paths with up to 2 edge-traversals.
• 2 - Enrichment in $$W$$ could capture some sort of notion of throughput.. If we imagine it as plumbing, given a water pipe $$\mathcal{X}(a, b)$$ and a water pipe $$\mathcal{X}(b, c)$$ the water pipe $$\mathcal{X}(a, c)$$'s throughput is giv…
• [ \begin{array}{ c | c c c c } \nearrow & A & B & C & D \\ \hline A & 0 & \infty & 3 & \infty \\ B & 2 & 0 & \infty & 5 \\ C & \infty & 3 & 0 & \infty \\ D & \infty & \infty &a…
• [ \begin{array}{c|cccc} d(\nearrow) & A & B & C & D \\ \hline A & 0 & 6 & 3 & 11 \\ B & 2 & 0 & 5 & 5 \\ C & 5 & 3 & 0 & 8 \\ D & 11 & 9 & 6 & 0 \end{array} ]
• A Lawvere metric space is a set of objects between which we have some sort of notion of a distance. It can be infinite which intuitively means that no matter the effort, sometimes you can't get from a point $$a$$ to the point $$b$$. On the other…
• $$d(US, Spain)$$ is bigger because US is larger. It takes longer to cross the entire US than the entire of Spain, since we have to cross the sea in both cases and can pretty much ignore it.
• I found two symmetric monoidal structures on $$(\textrm{Prop}^{\mathbb{N}}, \leq)$$. One is [ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) ] and the other one is [ (\textrm{Prop}^{\mat…
• Monoidal unit is 1. The $$*$$ operation is associative and commutative, so it satisfies properties (c) and (d). Multiplication of any $$x \in \mathbb{N}$$ by 1 yields back $$x$$, so (b) holds as well. Multiplication by any \(z \in \mathbb{N}\…
• Using Proposition 1.93, we can check just 7 things: [ \begin{array}{c|c|c|c} p&f(p)&g(f(p))&p \leq g(f(p))\\ \hline 1 &1&1&T\\ 2 &2&2&T\\ 3.9&4&4&T\\ 4 &4&4&T\\ \end…
• @Matthew, sure, that makes sense. Perhaps it's better to leave all autodiff discussion for after the Compiling to Categories is implemented, so there is less clutter in the thread.
• Hi everyone, I'm very excited to see so many interesting things in this thread! I'd like to add one thing to the mix, which is my attempt at implementation of @Conal's "Simple essence of automatic differentiation" in Haskell, since I see you're talk…