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Fredrick Eisele

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  • There are only id morphisms in \( \mathcal{J} \) which is isomorphic to \( \textbf{2} \). [ V_{\mathcal{J}} = \lbrace v, w \rbrace \\ A_{\mathcal{J}} = \lbrace 1_v, 1_w \rbrace ] [ X \times Y = \lim_{\mathcal{J}} D := \lbrace (D(v), D(w)), (D(1_v),…
  • Puzzle 106. What are some reasonable equations between morphisms that we might want to impose? Maybe we want to limit the number of layers of management to say 4. [ \text{Manager} \circ \text{Manager} \circ \text{Manager} \circ \text{Manager} =…
  • The title says Chapter 2, it should be 3.
  • Sorry, what? Are you commenting on the content or the format?
  • How is #17 not a preorder. I presume that the unique category with a single object, \(\star\), and one morphism, \(1_{\star}\) is a preorder. Is it because \( f \neq 1_{\star} \)? Why does that matter?
  • Maybe the simplest category without a terminal object is \( \textbf{2} := [ 1 \, 2 ] \) the discrete category of two objects. Does \( \textbf{0} \) qualify? It has no morphisms (to any object) but it has no objects from which to draw a terminal-obj…
  • 3) The recursive function is [ f(n)= \begin{cases} 0, & \text{if $x \le 0$}.\\ f(n-1) + n, & \text{otherwise}. \end{cases} ] The new element has a morphism to each of its predecessors \(n - 1\) and one to itself. Which is equivalen…
  • 1) \( X \times B \rightarrow Y \times B \) 2) \( X^B \rightarrow Y^B \) 3) \( +(-,3) = p(3) \) where \( p(3): \mathbb{N} \rightarrow \mathbb{N} \) and which sends \( (b) \mapsto 3 + b \). Does \( (X \rightarrow Y) \times B = X \times B \rightarr…
  • @KeithEPeterson I thought I was restating your puzzle. Generating graphs that encode monotonically increasing functions on the natural numbers sounds interesting.
  • @KeithEPeterson I presume you mean... Puzzle 104+. Given an arbitrary connected graph having a node labeled 'x', how many paths of length 'n' go from 'x' to 'x' in that graph? Where the resulting function \( H: \mathbb{N} \rightarrow \mathbb{N} \) …
  • We pulled back I along G.
  • [ \textbf{Gr} \overset{G}{\rightarrow} \textbf{DDS} \overset{I}{\rightarrow} \textbf{Set} ] \( G.I : \textbf{Gr} \rightarrow \textbf{Set} \) [ \begin{matrix} \begin{array}{c | c c} \text{Arrow} & \text{source} & \text{target} \\ \hline 1 …
  • I had a similar confusion. Consider three functions \( L, C, R \) where \( C : A \rightarrow B \) and \( L, R : B \rightarrow A \). Where \( L \) and \( R \) are left and right adjoint to \( C \) respectively. This implies that \( C \) is left adj…
  • Let me take a shot. With reference to comment 1. Yes, \(D\) is a functor the category is \(\mathcal{J}\) Definition 3.41 A diagram in \(C\) is a functor from any category \( D : \mathcal{J} \rightarrow C \). We say that the diagram \(D\) commute…
  • Puzzle 101 We have a collection of sets which, due to unique arrows among themselves, necessarily forms a preorder. As the arrow between members of this 'subcategory' are unique they do not carry any additional information. The 'subcategory' has …
  • There is the potential for confusing morphisms with paths. (3.7) in Example 3.6 and Exercise 3.7 \( \mathcal{N} = \textbf{Free} ( \) \( ) \) Has many paths, \(\aleph_0\), each of which is a distinct morphism. If we add equations the number of mo…
  • I think the authors were trying to motivate Proposition 1.86 by their remark. As an alternative to the 'crossing' arrows idea is that of approaching from above [or as drawn from the right] by the right adjoint and from below [or as drawn from the le…
  • Consider Remark 1.78 where a claim is made about crossing arrows, if they are 'bending'. In the book it is not clear that the arrows are crossing, in the diagram above it is pretty clear that they cross. What does arrow crossing signify?
  • The graph instances are offset so you can more easily complete the exercise.
  • 1) Any \(\alpha_c\) in \( \mathcal{P} \) is unique, because \( \mathcal{P} \) is a preorder meaning at most 1 arrow exists between a pair of objects. 2) There is nothing in \( \mathcal{C} \) restricting the number of arrows between a pair of obj…
  • @JohnBaez said "Good users of LaTeX use something like \( \) and \[ \] rather than $ $ and $$ $$. For global search and replace, it's very useful to be able to easily tell what's a left parenthesis and what's a right parenthesis!"
  • This example has some nice features. The math statements are clearly centered and tagged with names. Justification for the next for statement is provided with appropriate embedded math statements. It uses the QED box It provides an edit note indi…
  • 1) The monogamous spousal function for all people, where unmarried people are treated as their own spouse. The spouse, \(s\), of my spouse, \(z_1\) is me, \(z_0\). 2) The a mother, \(a\), of only children, \(c\), where \(g\) is the firstborn and \…
  • @KeithEPeterson Could you copy your examples over to the https://forum.azimuthproject.org/categories/applied-category-theory-formula-examples
  • In Example we were finding all the elements of \( Ob(\textbf{Set}) \). Do they all qualify as functors here?
  • I presume that the person is making a statement about graphs generally and not a specific graph. The person is confused about the difference between a round-trip vs. a trip for which has a return flight.
  • The \( | \textbf{Set} (n, m) | = | m |^{|n|} \).
  • Apparently there is a difference between paths and morphisms. I believe the morphisms form an equivalence group over the paths. For example, to which morphisms do the following paths belong? $$ id_z $$ $$ s $$ $$ s.s $$ $$ s.id_z.s.s $$
  • I add the Exercises as I work them out myself. Generally I have not been providing my own answers. If I doubt my answer then I put in my best guess. The Chapter 2 exercises are now present and I am starting on Chapter 3.