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limit of a functor (Definition)

Let $ G$ be a functor from categories $ \mathcal{I}$ to $ \mathcal{C}$. A limit of $ G$ is a pair $ (L,\tau)$ where

  1. $ L:\mathcal{I}\to \mathcal{C}$ is a constant functor,
  2. $ \tau: L\to G$ is a natural transformation,
such that it is universal among all pairs satisfying (1) and (2). In other words, for any pair $ (M,\varphi)$, where $ M:\mathcal{I}\to \mathcal{C}$ is a constant functor, and $ \varphi: M\to G$ is a natural transformation, there is a unique natural transformation $ \phi:M\to L$ with the following commutative diagram:
$ \xymatrix@C+=64pt{ M\ar[dr]^{\varphi}\ar@{.>}[dd]_{\phi}\ &G\ L\ar[ur]_{\tau} } $

Since we may identify a constant functor with its value, say an object $ A$, in $ \mathcal{C}$, a limit of $ G$ may be viewed as an object $ A$ in $ \mathcal{C}$, together with a collection of morphisms $ A \to G(I)$, or $ G_I$, for each object $ I$ in $ \mathcal{I}$ such that

if $\displaystyle I\to J$ is a morphism, then $\displaystyle A\to G_I\to G_J = A \to G_J.$ $\displaystyle \qquad (*)$    

Furthermore, if another object $ B$ in $ \mathcal{C}$ satisfies $ (*)$, then there is a unique morphism $ B\to A$ such that
$\displaystyle B\to A\to G_I = B\to G_I$
for all objects $ I$ in $ \mathcal{I}$.

Remarks.

  • A limit of a functor may or may not exist. If it exists, then it is unique up to natural equivalence. In other words, we may call it the limit of $ G$.
  • In the literature, the limit of $ G$ is variously known as the inverse limit, left limit, projective limit, root, or left root of $ G$. and is generally written $ {\mathrm{\varprojlim}}G$.
  • The category $ \mathcal{I}$ above is usually called the index category, and the functor $ G$ a diagram in $ \mathcal{C}$.
  • The most common index categories are finite categories, discrete categories, partially ordered categories, and more specifically, directed categories.
  • On the other hand, it turns out that if, in a category, if equalizers of any pairs of morphisms, and arbitrary products of any collections of objects exist, then it can be shown that every functor into this category has a limit.

By reversing all the arrows in $ \mathcal{C}$, we arrive at the dual concept of a limit, that of a colimit. Precisely, the colimit of $ G:\mathcal{I}\to \mathcal{C}$ is a pair $ (R,\tau)$ where $ R:\mathcal{I}\to \mathcal{C}$ is a constant functor and $ \tau: G\to R$ is a natural transformation, such that for any pair $ (M,\varphi)$ of constant functor $ M:\mathcal{I}\to \mathcal{C}$ and natural transformation $ \varphi: G\to M$, there is a unique natural transformation $ \phi: R\to M$ such that the following diagram

$ \xymatrix@C+=64pt{ &R\ar@{.>}[dd]^{\phi}\ G\ar[ur]^{\tau} \ar[dr]_{\varphi}\ &M } $
The colimit is also known as the direct limit, right limit, inductive limit, coroot, or right root, and is written $ {\mathrm{\varinjlim}}$.

Examples. Some common examples of limits (inverse limits) are products, terminal objects, pullbacks, equalizers, kernels, and kernel pairs. These examples can be readily verified. Let us verify that a terminal object is a limit:

Let $ G$ be the empty functor (from the empty category) into an arbitrary category $ \mathcal{C}$. So the limit of $ G$ is just an object $ C$ in $ \mathcal{C}$, and that's it, as there are no objects in the empty category, there are no morphisms from $ C$ in the limit of $ G$. If $ A$ is any object in $ \mathcal{C}$, then there is a unique morphism $ A\to C$, and that's it. But this means that $ C$ is just a terminal object of $ \mathcal{C}$.

Please see the verification of some of these examples in the attachments below.

Some examples of colimits (direct limits) are coproducts, initial objects, pushouts, coequalizers, cokernels, and cokernel pairs.



"limit of a functor" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: category, limit, inverse limit, category associated to a partial order, representable functor, $\mathcal{U}$-small, Grothendieck category, limiting cone

Other names:  inverse limit, projective limit, left limit, left root, root, direct limit, inductive limit, right limit, right root, coroot
Also defines:  limit, colimit, index category

Attachments:
categorical direct product is an inverse limit (Theorem) by archibal
kernel is an inverse limit (Theorem) by mathcam
direct limit of sets (Example) by CWoo
equalizer is an inverse limit (Example) by CWoo
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Cross-references: cokernel pairs, cokernels, coequalizers, pushouts, initial objects, coproducts, empty category, empty functor, kernel pairs, kernels, pullbacks, terminal objects, products, equalizers, directed categories, partially ordered categories, discrete categories, finite, diagram, natural equivalence, morphisms, collection, object, commutative diagram, universal, natural transformation, constant functor, categories, functor
There are 85 references to this entry.

This is version 25 of limit of a functor, born on 2004-02-24, modified 2008-10-03.
Object id is 5619, canonical name is DirectLimit.
Accessed 18215 times total.

Classification:
AMS MSC: 18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits )
Pending Errata and Addenda
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