Definition of Limits in Category Theory

Question

I was reading Kashiwara, Schapira's book Categories and Sheaves, in that limit of a projective system,

$$P:\mathcal{I}^{\text{op}}\to \textbf{Sets}$$

Is defined as follows,

$$\lim P = \text{Hom}_{\textbf{Sets}^{\mathcal{I}^{\text{op}}}}(Pt, P)$$

where $Pt$ is the constant functor. Can someone tell me what's happening here? I know the standard universal property definition. This seems extremely abstract to me, and can't make sense of it.

Answer 1

The universal property definition of a limit (of a projective system, in this case) tells you what property an object (+ morphisms) must have to be that limit. It is a definition that works for every category.

Now, some categories 'have projective limits' and some don't and to show that a particular category 'has projective limits', you need to actually construct that object.

Now $$\text{Hom}_{\textbf{Sets}^{\mathcal{I}^{\text{op}}}}(Pt, P)$$ is a construction of the projective limit in the category of sets.

Here $Pt \colon {\cal I}^{\text{op}} \to \textbf{Sets}$ is the constant functor that always gives the one-point set. An element of $\text{Hom}_{\textbf{Sets}^{\mathcal{I}^{\text{op}}}}(Pt, P)$ consists of a family of morphisms $f_i \colon Pt(i) \to P(i)$, for $i \in {\mathcal I}$, that commute with the morphisms $P(i \to j) \colon P(j) \to P(i)$ and $Pt(i \to j) \colon Pt(j) \to Pt(i)$. Since $Pt(i)$ is just a one-point set, you can as well see that as elements $f_i \in P(i)$, for $i \in {\mathcal I}$, such that for every morphism $k \colon i \to j$ in ${\cal I}$, $P(k)(f_j) = f_i$. This looks more like the 'usual' construction of the projective limit in $\textbf{Sets}$ as a family of ${\cal I}$-indexed elements of $P(i)$ that map to each other under the morphisms $P(i \to j)$.

Of course, it needs a proof that this is indeed the projective limit in $\textbf{Sets}$.

Answer 2

For a moment, define $\lim P$ in the ordinary way as an element of $\mathbf{Set}$ with a cone $\lim P \Rightarrow P$ satisfying the appropriate universal property. Notice that the set $\textrm{Hom}_{\mathbf{Set}^{\mathcal{I}^{\textrm{op}}}}(Pt,P)$ is exactly the set of cones $t\Rightarrow P$. Because of the universal property, there will be a one-to-one correspondence between arrows $t\to \lim P$ and cones $t\Rightarrow P,$ giving a natural isomorphism $$ \textrm{Hom}_{\mathbf{Set}}(t,\lim P) \cong \textrm{Hom}_{\mathbf{Sets}^{\mathcal{I}^{\textrm{op}}}}(Pt,P). $$ It can be proven (with the Yoneda Lemma) that limits of $P$ are exactly objects satisfying the above identity, and that a choice of natural isomorphism completely specifies a universal cone $\lim P\Rightarrow P.$ Restating this fact, limits of $P$ are exactly objects that represent the contravariant functor $\textrm{Hom}_{\mathbf{Sets}^{\mathcal{I}^{\textrm{op}}}}(P-,P):\mathbf{Set}^{\textrm{op}}\to \mathbf{Set}.$ Kashiwara and Shapira then make the choice to conflate the limit with the functor that it represents.

Edit: After looking at Kashiwara and Schapira's book I realized that Magdiragdag's answer is actually much closer to what they are going for. My mistake was to assume that they understand $t$ to be a free variable, when actually it is meant to specifically be a one element set. Fixing $t$ as a one element set we get $$ \lim P\cong \textrm{Hom}_{\mathbf{Set}}(t,\lim P)\cong \textrm{Hom}_{\mathbf{Sets}^{\mathcal{I}^{\textrm{op}}}}(Pt,P), $$ so that Kashiwara and Schapira are in fact defining the limit as a set rather than a contravariant functor. Quickly re-iterating the response by Magdirag, the limit can be identified with the set of all tuples $(f_i)_{i\in \mathcal{I}}$ that are compatible under the morphisms induced by $P$. The universal arrows are given by the projection maps onto the components of the tuple. This set can then be identified with the set of cones $\textrm{Hom}_{\mathbf{Set}^{\mathcal{I}^{\textrm{op}}}}(Pt,P).$

Answer 3

Here is how this can be shown in a more mechanical way: $$ \mathrm{lim}\,P \cong \mathbf{Sets}(\mathbb{1}, \mathrm{lim} P) \cong \mathbf{Sets}^{\mathcal{I}^\mathrm{op}}(\Delta\mathbb{1}, P) $$

Here are the two patterns, which can be easily applied almost without paying any attention to the surrounding text:

• $A \cong \mathbf{Sets}(\mathbb{1}, A)$, because sets are determined by their elements
• $\mathcal{C}^\mathcal{J}(\Delta A, D) \cong \mathcal{C}(A, \mathrm{lim}\,D)$, because $\Delta \dashv \mathrm{lim}$ (where $\Delta: \mathcal{C} \to \mathcal{C}^{\mathcal{J}}$ is the diagonal functor).

The difference to what subrosar wrote might seem subtle, but simply replacing $Pt$ by $\Delta \mathbb{1}$ makes the connection to the catchy

$$ \mathrm{colim} \dashv \Delta \dashv \mathrm{lim} $$

immediately apparent.

Of course, one also has to remember why "$\Delta \dashv \mathrm{lim}$" is exactly the same as the universal-property-definition-spelled-out-in-prose-with-cones-and-indexed-families-of-morphisms. But once one has done that, it's much easier to remember "$\Delta \dashv \mathrm{lim}$", which is also arguably more convenient for quickly scribbling down some formulas.