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October 7

Leinster seems to be saying that proof of Yoneda lemma is a priori, but what about with in the relaxed category?

For example, in the semigroupoid, ${\hat {\alpha }}=\alpha _{A}(1_{A})$ is not guaranteed. Also, I don't think naturality holds for multiplicative graph. (because Coppey says that: square diagrams in a multiplicative graph can take one of five types depending on which potential compositions in the diagram are actually defined.) Will these examples useful for understand the Yoneda lemma? (I would suggest adding these two examples to the article.) What is the minimal condition (structure) under which the Yoneda lemma holds?--SilverMatsu (talk) 02:33, 7 October 2026 (UTC)

You would need to formulate a version of the Yoneda lemma that is meaningful in the relaxed context, before it can even be discussed whether it is true, and if so whether the proof is analogous. The usual formulation of the Yoneda lemma involves a family of hom functors, each denoted, for a given object $A,$ as $\mathrm {Hom} (A,-)$ or $h_{A}(-),$ defined on an arrow $f:P\to Q$ by $h_{A}(f)(g)=f\circ g,$ where $g:A\to P.$ For a multiplicative graph, $f\circ g$ is in general not defined, so it is dubious that a construction can be defined that assumes the role of these hom functors.  ​‑‑Lambiam 18:53, 7 October 2026 (UTC)

Thank you for your reply. I agree with you that $\mathrm {Hom} (A,f)$ is not guaranteed in a multiplicative graph. I'll looking into how to the define of the hom functor in a multiplicative graph. Also, since the composition of natural transformation is general not defined in a multiplicative graph, I think $\theta \circ -$ ($\theta \circ \alpha$) is not guaranteed.--SilverMatsu (talk) 03:48, 8 October 2026 (UTC)

Since composition is not guaranteed in a multiplicative graph, I looked for the definitions of the dual category and covariant/contravariant functors in multiplicative graph. Ehresmann says that: the dual class (classe multiplicative duale) of multiplicative graph is multiplicative graph. While, hom class is defined against the ordinary category. Since there isn't a electronic version, I copied it to my sandbox.--SilverMatsu (talk) 02:39, 10 October 2026 (UTC)

Reference

• Tom Leinster (2016). "Basic Category Theory". arXiv:1612.09375 [math.CT].

October 12

Why is a vertex with deg⁻(v) = 0 (vertex of a directed graph with no exiting edge) called a sink?

After looking at the Example 3.2.8. (atomic graph), I got what sink means, but I didn't understand why it's called a sink. This is a term from graph theory, but I suggest adding it to the Glossary of category theory, because it is used in the definition of the operation of pasting. It seems that sink isn't just another name for codomain (target).--SilverMatsu (talk) 02:57, 12 October 2026 (UTC)

Think of a faucet (source) and a drain (sink)? —Antonissimo (talk) 04:23, 12 October 2026 (UTC)

"Sink" has many meanings describing something where things disappear. Examples (per Wiktionary:sink): sink for water, heat sink, resource sink. I've used the phrase "time sink" to any describe a time consuming activity that produces little or no tangible benefit. In any case, definitions 10 & 11 cover the graph theoretical meanings, and I think the analogy is pretty clear. --RDBury (talk) 16:09, 12 October 2026 (UTC)

The article Pasting diagram is not comprehensible in its current state, since it uses the notion of "pasting diagram" throughout without ever defining it. It contains a definition of "2-pasting diagram", after the term "(pasting) diagram" has already been used 10 times, but this definition requires an understanding of the concept of "admitting a pasting scheme presentation", which also lacks a definition.

The term sink as used by Johnson & Yau is ancillary to their definition of an "anchored" (interior) face of a planar connected graph. It is not a category-theoretical term. In standard graph-theoretical parlance, the term is used in directed graphs, but Johnson & Yau use it to induce a direction on paths in an undirected graph, which is not standard at all.  ​‑‑Lambiam 07:40, 12 October 2026 (UTC)

By the way, where do you find Example 3.2.8.? —Antonissimo (talk) 17:41, 12 October 2026 (UTC)

In Johnson, Niles; Yau, Donald (2021). 2-Dimensional Categories. p. 74. arXiv:2002.06055, referenced below as well as in Pasting diagram.  ​‑‑Lambiam 21:52, 12 October 2026 (UTC)

Reference

• Johnson, Niles; Yau, Donald (31 January 2021). 2-Dimensional Categories. Oxford University Press. arXiv:2002.06055. ISBN 978-0-19-264567-8.