Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to prove Lemma 2.1.1 of Kashiwara-Schapira "Categories and Sheaves":

Let $\beta:I^{\text{op}}\to\mathbf{Set}$ be a functor ($I$ a small category) and let $X\in\mathbf{Set}$. There is a natural isomorphism $$\text{Hom}_{\mathbf{Set}}(X,\lim_{\leftarrow} \beta)\cong \lim_{\leftarrow}\text{Hom}_{\mathbf{Set}}(X,\beta)$$ where $\text{Hom}_{\mathbf{Set}}(X,\beta)$ denotes the functor $I^{\text{op}}\to\mathbf{Set}$ , $i\mapsto \text{Hom}_{\mathbf{Set}}(X,\beta(i))$.

This comes just after they have proved that $$\lim_{\leftarrow} \beta\cong \lbrace \lbrace x_i\rbrace_i \in\prod_i \beta(i)\;|\;\beta(s)(x_j)=x_i\;\text{for all}\; s:i\to j\;\text{in}\;I\rbrace$$ and they say that the lemma "is obvious". Why is it obvious?

(By the way, why are they using the notation of a projective limit? They are defining a general limit in the categorical sense, right?)

share|cite|improve this question
Terminological: this is not a limit in a presheaf category, this is a limit of a presheaf in the category $\mathbf{Set}$. – Oskar Feb 17 '14 at 22:05
"Projective limit" is their terminology for what is normally called a "limit". Their "inductive limit" is what is normally called a "colimit". – Uday Reddy Feb 18 '14 at 18:56

Remember the universal property of a limit: the maps from $X$ to $\varprojlim \beta$ are in one-to-one correspondence with the cones over $\beta$ with tip $X$. One builds such cones out of maps from $X$ to $\beta(i)$ which commute with the images of morphisms in $I$ under $\beta$. More precisely, such a cone is specified as an element of $\prod_I \hom(X,\beta(i))$ satisfying the same relations as in the explicit definition you give of a limit. This is just a useful and simple rephrasing of the definition of limit, which can easily get cloaked in technicalities.

The projective limit notation is somewhat standard for an arbitrary limit: projective limits are special cases of limits, and inductive limits of colimits, so it's natural to write $\varprojlim$ for a limit and $\varinjlim$ for a colimit. The alternative is to write $\lim$ and $\text{colim}$. Kashiwara-Schapira is generally considered to be pretty difficult reading, incidentally-if you're new to category theory you might spend some time with a more elementary text alongside.

share|cite|improve this answer
Yes, but they haven't defined the limit via a universal property. They have just defined it by setting $$\lim_{\leftarrow} \beta=\text{Hom}_{[I^{\text{op}},\mathbf{Set}]}(1,\beta)$$ – triwer23 Feb 17 '14 at 21:24
OK, then a map from $X$ to $\varprojlim \beta$ is a choice for each $x\in X$ of a natural transformation from $1$ to $\beta,$ i.e. a point in $\beta(i)$ for every $i$ commuting with maps in $I$. This gives a function from $X$ to $\beta(i)$ for every $i$ commuting with maps in $I$, which is what a cone is. – Kevin Carlson Feb 17 '14 at 21:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.