能否用Cat上函子的不动点刻画泛构造及相关范畴性质？

Great question—this ties together two core ideas in category theory: universal constructions and functorial fixed points (in the equivalence sense). Let’s unpack both parts clearly:

1. Can universal constructions be described as fixed points of functors on Cat? #

Absolutely! The intuition here is that a universal construction (like products, terminal objects, or exponentials) is a way of "closing" a category under a specific operation. When a category already has all instances of that universal structure, applying the corresponding "completion functor" to it will yield a category equivalent to the original—meaning the original category is a fixed point (up to equivalence) of that functor.

For example, if you have a functor that takes a category and adds a terminal object (if it doesn’t already have one), then categories with terminal objects are exactly the fixed points (up to $\simeq$) of this functor. The same logic extends to products, coproducts, exponentials, and more.

2. Can we characterize category properties via $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$? #

Yes! For each of the properties you mentioned, we can define a specific functor $\mathcal{F}: Cat \to Cat$ such that the property holds if and only if $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$. Here are concrete, actionable examples:

Terminal Objects #

Define $\mathcal{F}(\mathcal{C})$ as the category formed by extending $\mathcal{C}$ with:

• A new object $t$
• The identity morphism $id_t$ on $t$
• A unique morphism $X \to t$ for every object $X$ in $\mathcal{C}$
• All original morphisms and composition rules from $\mathcal{C}$

Then $\mathcal{C}$ has a terminal object if and only if $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$. Why? If $\mathcal{C}$ already has a terminal object $t_0$, then $t$ in $\mathcal{F}(\mathcal{C})$ is isomorphic to $t_0$, making the two categories equivalent. Conversely, if $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$, the image of $t$ under the equivalence gives a valid terminal object in $\mathcal{C}$.

Binary Products #

Define $\mathcal{F}(\mathcal{C})$ as the category that extends $\mathcal{C}$ by adding, for every pair of objects $A,B \in \mathcal{C}$:

• A product object $A \times B$
• Projection morphisms $\pi_A: A \times B \to A$ and $\pi_B: A \times B \to B$
• For every pair of morphisms $f: X \to A$, $g: X \to B$, the unique morphism $\langle f,g \rangle: X \to A \times B$ satisfying $\pi_A \circ \langle f,g \rangle = f$ and $\pi_B \circ \langle f,g \rangle = g$
• All necessary composition rules to maintain categorical structure

Then $\mathcal{C}$ has all binary products if and only if $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$. If $\mathcal{C}$ already has all binary products, the new product objects added by $\mathcal{F}$ are isomorphic to existing ones in $\mathcal{C}$, so the categories are equivalent. If $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$, the equivalence maps the new product objects to valid products in $\mathcal{C}$.

Cartesian Closed Categories (CCCs) #

This is more complex, as CCCs require two key features: all binary products, and for every object $A$, the functor $- \times A: \mathcal{C} \to \mathcal{C}$ has a right adjoint (the exponential functor $(-)^A$).

Define $\mathcal{F}(\mathcal{C})$ as the completion of $\mathcal{C}$ that:

1. First adds all missing binary products (following the binary product functor above)
2. Then adds, for every pair of objects $A,B \in \mathcal{C}$:
   • An exponential object $B^A$
   • The evaluation morphism $eval: B^A \times A \to B$
   • For every morphism $f: X \times A \to B$, the unique curried morphism $\lambda f: X \to B^A$ satisfying $eval \circ (\lambda f \times id_A) = f$

Then $\mathcal{C}$ is a Cartesian closed category if and only if $\mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$. The equivalence confirms $\mathcal{C}$ already has all required products and exponentials, satisfying the CCC axioms.

A few quick notes to clarify:

• These functors $\mathcal{F}$ are typically large functors (since they can add objects/morphisms to small categories), but they still map into $Cat$ (the category of small categories and functors) when working with equivalence.
• We use categorical equivalence $\simeq$ instead of strict equality because universal objects are only unique up to isomorphism—this is the natural, standard notion of "sameness" in category theory.

内容的提问来源于stack exchange，提问作者Nathan BeDell