# Definition of Limits in Category Theory

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.

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 $$\textbf{Sets}$$.

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 $$\textrm{Hom}_{\mathbf{Set}^{\mathcal{I}^{\textrm{op}}}}(Pt,P).$$

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:

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.