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created an entry category of being, for completeness.
A general remark: people often write that, unfortunately or not, Eilenberg-MacLane’s term “category” is not that of, say, Kant. But in fact if read this way here, following Lawvere, then the former is a good formalization of the latter, after all.
Care to provide a link to the MO question? Did it get deleted?
Fair enough. Not sure how much traction you’ll get here, either, since the number of people who can pass Hegelian thoughts through hopefully the clarifying lens of Lawvere’s proposed interpretation is rather small. And those few people might be busy with something else.
The answer to the third question is expanded on at Aufhebung of Becoming
Regarding the first question: All the other objects is what appears in between as the two opposite extremes of the initial opposition are pulled apart:
$\varnothing \longrightarrow X \longrightarrow \ast$.
This being a special case of the general way in which adjoint modalities exhibit all objects as equipped with two opposite properties, see here.
Hi Dean,
(1) The general rule is that types are objects and terms are morphisms. In intrinsic type theories, terms are uniquely typed, so one cannot have $a: A$ then also $a:B$ in a non-definitionally equivalent type. This section of Science of Logic addresses Aristotelian syllogistic logic. In the case of $f: A \to B$, glossed as ’All $A$ are $B$’, as it says latter there, often one works with a context, $C$. If then $A = \sum_{x:C} P(x)$ and $B = \sum_{x: C} Q(x)$, for predicates $P(x)$ and $Q(x)$, then $f$ might arise from an implication $P(x) \to Q(x)$. We’re perhaps inclined here to speak of a $c: C$ such that $P(c)$ and $Q(c)$ is true, and then have $c: A$ and $c:B$, but this isn’t strictly correct.
There’s a question of how the translation works from Hegel to Lawvere and to nLab, whether and how one requires some kind of agreement with the spirit of the writing of someone who lacked any formal means to say things we now can. But it may be easier to approach from the other side and take what’s in Lawvere and nLab as rather interesting, and reasonably inspired by Hegel. The portion on Being and Nothing begins here. As Urs writes, the adjoint modalities $\emptyset \dashv \ast$ is at the heart of this passage, expressing the unity of two moments. You can think of it generated by the adjoint triple, formed of left and right adjoints to the terminal functor $\mathbf{H} \to \ast$, so that $0$ and $1$ are distinct images of $\ast$ (full subcategories of $\mathbf{H}$).
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