Note that we are not implying that category theory is preferable to set theory as a foundation for mathematics or even for science in general. We are inclined to hold that dependent type theory is a good candidate for such a role.
From a categorical point of view what is the simplest object we can conceive ? The singleton category with only one object $\star$ and only one arrow, the identity morphism $id_{\star}$ on this object. All such singleton categories are equivalent and a singleton category is in fact the terminal object in the $2$-category $Cat$ of small categories. They are complementary because they emerge from right and left adjoints respectively (which are necessarily fully faithful) of the same unique functor $\mathbb{T}: \mathcal{C}\rightarrow \{\star\}$.
The first section of Hegel's logic, the logic of being, involves an abstract exploration of the concept of 'space' and the differential geometry used in mathematical physics. Hegel, who taught differential calculus both at high school and university, dedicates a long section of the Logic to the infinitesimal calculus, the notions of which illuminate many other passages of the Logic. We start with what can be considered the archetype of the concept of space, the adjunct quadruple associated to a the category of presheaves over a category $C$ with a terminal object. We call this space proto-space.
Here $\Gamma(P) = P(1)$. It is very instructive to work out these adjoints $\Pi\vdash Disc \vdash\Gamma\vdash CoDisc$ explicitly. We have that $Disc (X)(A) = X$ for all objects $A$ in $C$ and $Disc(X)(f)$ is $id_X$ for any $ f : A \rightarrow B$ in $C$. We have that $\Pi (\mathcal{A}) := \bigcup_{U \in Obj C} \mathcal{A}(U) / \sim$ where $\sim$ is the equivalence relation which identifies $s$ and $s'$ over $U$ and $U'$ respectively if there are $f: V \rightarrow U$, $g : V \rightarrow U'$ with $s_V = s'_V$. If $\Omega$ is the subobject classifier for presheaves on a topological space $X$ then $\Pi(\Omega)$ gives the connected components of $X$ in the usual sense. We have finally that $CoDisc(X)(U) = \mathcal{P}X$. If we have a set map $f : \mathcal{A}(1) \rightarrow X$ then we
can define a morphisms of presheaves $f^\flat : \mathcal{A} \rightarrow Codisc(X)$ given by $f^\flat (U)(s)= \{ x \in X : \exists w \in \mathcal{A}(1), w_U = s \}$ for $s \in \mathcal{A}(U)$.
This quadruple is closely related to the above diagram for terminal and initial objects. The functors in the quadruple arise by taking the left and right Kan extensions along $\mathbb{T}$ and $1$. Thus we have the development of the concept of proto-space from that of being.
The terminal object is the limit of the empty diagram. (Co)limits and adjunctions are all special cases of Kan extensions. Thus starting from the 2-category of all categories we can derive the concept of proto-space employing only Kan extensions. Kan extensions express dialectical reason's process of passing to the other while preserving an essential mediating connection. Proto-space then assumes various form through various localisations giving rise to the sheaf toposes.
In general the adjoint quadruple does not carry over to a topos obtained by localisation. The condition of being cohesive is what guarantees this. One example of a cohesive topos are sheaves on a cohesive site. Thus we are lead spontaneously to the concept of cohesive topos as the right categorical notion of space.
But it should be stated that $Disc$ represents to movement towards discrete quantity or repulsion of units and $CoDisc$ represents the movement towards continuous quantity and coalescence of units.
The Yoneda embedding $C \rightarrow PrShv(C)$ expresses that each category $C$ unfolds into a proto-space. This unfolding of categories in the category of Being proceeds to something extrinsic, as a passage to the other. Dialectical reasoning asks for how something is constructed, for repressed history. A paradigm is that we can have diagrams into the category without the corresponding (co)limit existing.
But to think of a diagram and the concept of limit is already to posit the limit as something other and lacking in the category but nevertheless proceeding from it. Thus the complete and cocomplete category of presheaves given by the Yoneda embedding is a genuine Hegelian progression.
The key to understanding higher category theory is passing the above considerations into the correct generality of enriched category theory. A cosmos is what categories are enriched in. The above is the special case for the cosmos Set. Thus we should think of enriched presheaves, the enriched functor category between an enriched category and the cosmos itself seen as an enriched category.
Homotopy is the passage of quantity into quality. It is a changing of shape and size which preserves and thus defines a certain quality. Model categories are simply categorical abstractions where all constructions in classical homotopy theory can be carried out. The quality associated to a variation in quantity is expressed as the localisation yielding the homotopy category.
A simplicial set is an abstraction of a topological space, it is a categorical abstraction of a geometric form (i.e. a polytope). But it is a geometric form which contains within itself the process of its own genesis or assemblage, analogous to G-code (cf. the geometric realisation functor associating a topological space to each simplicial set). Category theory enriched over the cosmos of simplicial sets is currently seen to be the correct choice for the doing homotopy theory and differential geometry at the highest level of abstraction.
We must investigate the deeper significance of the simplex category $\Delta$ and its associated augmented simplex category $\Delta_a$ used to define simplicial sets. $\Delta_a$ has the natural structure of a strict monoidal category and $[0,1]$ has a natural monoid structure. This situation is universal in that monoidal categories $B$ with monoid objects $M$ are classified by functors $\Delta_a \rightarrow B$ sending $[0,1]$ to $M$. Similary we can obtain a classification of monads in a 2-category (curiously enough equivalent lax functors from the terminal object category). Perhaps $\Delta_a$ can be viewed as a higher qualitative categorical determination of the natural numbers expressing the exteriorisation and unfolding of the unit, one, monas (Hegel includes a short digression on Pythagoreanism in the section on Quantity).
$Set$ is an exterior, abstract, discrete concept (Hegel's concept of number seems to be very set theoretic), but the category of simplicial sets $sSet$ represents a greater cohesion between parts and qualitative structure and determination. The morphisms between objects in a simplicially enriched category instead of being a mere set become a space.
For a good introduction to higher topos theory see
https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes
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