Limit - endofunctor.dev

A **limit** of a [[Diagrams |diagram]] $D : \mathcal{J} \to \mathcal{C}$ is a [[Cones |cone]] $\lambda_j : \lim D \;\longrightarrow\; D(j) \quad (j \in \mathcal{J})$ with apex $\lim D$ that is **universal** among all cones to $D$. Concretely, this means: 1. It is a cone: $D(\alpha)\circ \lambda_j = \lambda_{j'} \quad\text{for every } \alpha: j \to j'$. 2. **Universal property**: For **any** other cone $\{\pi_j : N \to D(j)\}$, there exists a **unique** morphism $u \;:\; N \;\longrightarrow\; \lim D$ making all relevant triangles commute: $\pi_j \;=\; \lambda_j \,\circ\, u \quad \text{for all } j \in \mathcal{J}$. Diagrammatically, one depicts this as follows:  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNCxbMSwwLCJOIl0sWzEsMSwiXFxsaW0gRCJdLFswLDIsIkQoaikiXSxbMiwyLCJEKGonKSJdLFswLDEsIlxcZXhpc3RzIVxcLHUiLDEseyJsYWJlbF9wb3NpdGlvbiI6MjAsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDIsIlxccGlfaiIsMV0sWzAsMywiXFxwaV97aid9IiwxXSxbMSwyLCJcXGxhbWJkYV9qIiwxXSxbMSwzLCJcXGxhbWJkYV97aid9IiwxXSxbMiwzLCJEKFxcYWxwaGEpIiwyXV0=&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> - The bottom portion shows that $\{\lambda_j\}$ is itself a cone: $λj′=D(α)∘λj\lambda_{j'} = D(\alpha)\circ \lambda_j$. - The dashed arrow $u : N \to \lim D$ is the **unique** factor through which every other cone $\{\pi_j\}$ factors. If $\lim D$ exists, it is unique **up to unique isomorphism**. --- ## 2. Relationship to Cones and Diagrams By definition: - A **limit** of $D$ **is** precisely a “universal cone” to $D$. - Every cone to $D$ must factor through the limit cone (if the limit exists). Hence, the data of a limit is: an object (the apex) plus a universal system of maps into the diagram that is initial/terminal among all such cones (terminal in the slice sense, but “universal” among cones). --- ## 3. Limit and Equalizers ### Equalizers as Limits An **equalizer** of two parallel morphisms $f, g : X \rightrightarrows Y$ is the limit of a “parallel pair” diagram  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNCxbMSwwLCJFIl0sWzAsMSwiWiJdLFsxLDEsIlgiXSxbMiwxLCJZIl0sWzAsMiwiZSJdLFsxLDAsIntcXGV4aXN0cyF9IiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMiwieiIsMl0sWzIsMywiZiIsMCx7Im9mZnNldCI6LTF9XSxbMiwzLCJnIiwyLHsib2Zmc2V0IjoxfV1d&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> Namely, a limit of this diagram $D$ is exactly the object $E$ with a morphism $e: E \to X$ universal among all objects $Z$ that co-equalize $f$ and $g$. Concretely, “limit of the parallel pair” = “equalizer.” Thus **equalizers** are a special case of **limits**. --- ## 4. Limit and Pullbacks ### Pullbacks as Limits A **pullback** of a cospan $A \xrightarrow{\,f\,} C \xleftarrow{\,g\,} B$ is the limit of this functor:  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNSxbMSwzLCJBIl0sWzMsMywiQyJdLFszLDEsIkIiXSxbMSwxLCJBIFxcdGltZXNfQyBCIl0sWzAsMCwiWiJdLFsyLDEsImciLDJdLFswLDEsImYiXSxbMywwLCJcXHBpX0EiXSxbNCwyLCJ6X2IiLDAseyJjdXJ2ZSI6LTJ9XSxbNCwwLCJ6X0EiLDAseyJjdXJ2ZSI6Mn1dLFs0LDMsIlxcZXhpc3RzICEiXSxbMywyLCJcXHBpX0IiLDJdXQ==&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> Hence, **pullbacks** are also a special case of **limits** (the limit of the “cospan shape”). --- ## 5. Limit and Right Adjoints ### Right Adjoints Preserve Limits A well-known theorem states: > **Right adjoint functors** preserve all limits. That is, if $F : \mathcal{C} \to \mathcal{D}$ is **right adjoint** to some $G$, and if $\lim D$ is a limit in $\mathcal{C}$, then $F(\lim D)$ is (up to isomorphism) the limit of $F \circ D$ in $\mathcal{D}$. ### Limits as Right Adjoint to the Diagonal Functor On the other hand, the very existence of **limits** of a given shape $\mathcal{J}$ in $\mathcal{C}$ can be phrased as: $\Delta : \mathcal{C} \;\longrightarrow\; \mathcal{C}^\mathcal{J}$ (the **diagonal functor**, sending an object $C$ to the constant diagram at $C$) has a **right adjoint** $\mathrm{Lim} : \mathcal{C}^\mathcal{J} \to \mathcal{C}$. In that sense, a category having **all limits** of shape $\mathcal{J}$ is equivalent to the existence of a right adjoint to the diagonal functor. --- ## 6. Limit and Right Kan Extensions Any **limit** can be seen as a **right Kan extension** along the unique functor from $\mathcal{J}$ to the terminal category $1$. Concretely: - Let $D: \mathcal{J} \to \mathcal{C}$. - Consider the functor $D$ as a functor $\mathcal{J} \to \mathcal{C}$ that we want to “extend” along the unique functor $\mathcal{J} \to 1$. - The **right Kan extension** $\mathrm{Ran}_{\mathcal{J}\to 1}(D)$ is precisely the **limit** of $D$. Equivalently, every limit is a right Kan extension along that trivial functor $\mathcal{J} \to 1$. This conceptual perspective is very common in enriched / 2-category approaches to category theory. --- ## 7. Table: Limits of Various Diagram Shapes Many standard categorical constructions are particular instances of limits (or colimits, in the dual setting). Below is a short table of **examples of limits**: | **Construction** | **Diagram Shape** | **Universal Property** | | --------------------------------------------------- | ----------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------- | | **Terminal Object** | **Empty diagram** (no objects, no morphisms) | A limit over an empty diagram is an object $1$ such that there is a unique map $X \to 1$ for every object $X$. | | **Product** $(X \times Y)$ | **Discrete diagram of two objects** $X, Y$, no morphisms between them | A limit over this diagram is an object $X\times Y$ with projections satisfying the usual universal factorization. | | **Pullback** | **Cospan** $A \to C \leftarrow B$ | A limit over a cospan is an object $P$ with maps to $A$ and $B$ making a commuting square over $C$, universally. | | **Equalizer** | **Parallel pair** $X \rightrightarrows Y$ | A limit over two parallel arrows is an object $E\to X$ that “equalizes” ff and $g$, universally. | | **Inverse/Projective Limit** (of an inverse system) | **Any small category** $\mathcal{J}$ with “directed” or “partial order” shape | The limit of a functor $D : \mathcal{J} \to \mathcal{C}$ generalizes all the above, capturing infinite cones as well. | - **Terminal object** $\leftrightsquigarrow$ limit over the “empty shape.” - **Equalizers**, **pullbacks**, **products**, etc. $\leftrightsquigarrow$ limits over their respective shapes. - **All small limits** $\leftrightsquigarrow$ right adjoint $\mathrm{Lim}$ to diagonal $\Delta : \mathcal{C}\to\mathcal{C}^\mathcal{J}$. --- ## Concluding Summary 1. A **limit** of a diagram $D: \mathcal{J}\to\mathcal{C}$ is a **universal cone** $\lim D$ over $D$. 2. **Equalizers** (limit of two parallel arrows) and **pullbacks** (limit of a cospan) are classical, concrete examples of limits. 3. A functor $F$ that is a **right adjoint** preserves all limits (including equalizers, pullbacks, etc.). 4. A **right Kan extension** perspective: all limits arise as right Kan extensions of a diagram to the terminal category. 5. Many important categorical constructions are **particular instances** of limits of specific shapes. A category that has all these shapes’ limits is called **complete** (for _all_ small shapes). This unifying viewpoint is one of the great strengths of category theory: many classical constructions in mathematics become special cases of “limit of a diagram.”