#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

# Cheuk Man Hwang

• I am trying to see if we can interpret an $$n$$ by $$m$$ real matrix $$\mathbf{A}$$ as a functor so that its conjugate transpose $$\mathbf{A}^*$$ is the right adjoint in the categorical sense and use it to understand the meaning of naturality. Howe…
• Alright, I'm thinking out loud here, trying to cook up the functor $$\mathbf{Cat}\overset{R_1}\longrightarrow\mathbf{Preord}$$ in Puzzle 163. Currently I am only thinking about how $$R_1$$ acts on the objects. So let's pick a random category $$\ma… • @Keith Like I said, I suspect that John left out some technicality so that we can focus on the essential features of category theory. To be fully precise, we should define \(\mathbf{Cat}$$ as the category of all small categories, which means the me…
• @Keith comment38, the collection of objects $$\mathrm{Ob}(\mathbf{Set})$$ in the category $$\mathbf{Set}$$ is not a set, it is a class. This means $$\mathbf{Set}$$ is a large category, echoing Anindya's comment#41. This is a way to resolve Cantor'…
• Elaborate a bit more on Keith's answer. Puzzle 161 1) Preservation of composition: Suppose $$h\in\mathcal{C}(c, c')$$ and $$(f,g)$$ is a morphism from $$(c, c')$$ to $$(d, d')$$ and $$(l, j)$$ is a morphism from $$(d, d')$$ to $$(e, e')$$, we hav…
• Puzzle 136 I understand Jonathan and Matthew had already answered this puzzle in comment 2 and comment 32, but I would like to fill in some details. As Jonathan pointed out, functors from the category $$\mathbf{m}$$ to the category $$\mathbf{n}$$ …
• John wrote: Puzzle. If p is prime, how many actions of the group Z/p are there on a set with n elements? For a database with schema $$\mathbb{Z}_p$$ and maps the node to a set with n elements, the edge is mapped to either the identity or an n-…
• Puzzle 85 Suppose $$y\in\mathbb{N}(T)$$, hence $$y=a_{E}[E]+a_{Y}[Y]+a_{W}[W]$$ where $$a_E, a_Y, a_W\in \mathbb{N}$$. I am not sure if I am correct but my observation is that in $$\mathbb{N}(T)$$, even the reflexive property implies that a_{E}…
• The monoidal unit, call it $$I$$, should be $$false$$ because $$false\vee I = I\vee false= false \vee false = false$$ $$true\vee I = true \vee false =true = false\vee true = I\vee true$$ Moreover, \((false\vee false)\vee false=false=false\vee(…