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• CommentRowNumber1.
• CommentAuthorDexter Chua
• CommentTimeSep 23rd 2016
• (edited Sep 23rd 2016)

Added some more explicit constructions in limits and colimits by example, which are hopefully more pedagogically appropriate, as Urs suggested in the discussion. A lot more details are to be filled in.

• CommentRowNumber2.
• CommentAuthorDylan Wilson
• CommentTimeSep 23rd 2016

One fun thing would be to show by explicit example how each of the colimits in Set can be computed as a limit. For example: the empty set is one of the elements in the set of subsets of the one element set, quotients by an equivalence relation are modeled by the set of cosets, and coproducts over an index set A can be modeled in the following way: take the product of all the power sets and look at those A-tuples which are the empty set everywhere except one place where it’s a singleton. Then add a link to ’exponential object’ and ’topos’ or whatever…

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeSep 24th 2016

…as a limit involving power objects. Since the barest definition of elementary topos doesn’t mention colimits (power objects and finite limits only), all existing colimits can be built using these (only finite, or all, depending on taste).

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeSep 24th 2016

Would it be out of place to mention this work in progress, indicating how to prove this fact about an elementary topos using reasoning closer to the naive set theory? An Elementary Approach to Elementary Topos Theory. There are still some parts to fill in, but I’ve put it aside for a few months. May get back to it soon.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeSep 24th 2016

I was thinking of your old write-up Todd, so I’m glad this has progressed so much more! I seriously think this should eventually be published in a more traditional manner.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeSep 25th 2016

1. At limits and colimits by example, there was an old query box by Mark C. His observation was right and I changed the entry accordingly.

In the section on general colimits, under colimits, it stated that

The equivalence relation $\sim$ is that which is generated by

$((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists (f : d \to d') with F(f)(x) = x') \,.$

If $D$ is a filtered category then the relation $\sim$ already is an equivalence relation.

Is this true? Consider a category with two objects $d, d'$ and a single non-identity morphism $f: d \rightarrow d'$. Then this is a directed category.

Consider the functor $F: D \rightarrow Set$ taking both $d$ and $d'$ to singletons.

Then, it seems to me the unique $x \in F(d)$ would be related to the unique $x' \in F(d')$, but the opposite relation would not hold, since there is no morphism $d' \rightarrow d$ in $D$. Hence symmetry fails and this is not an equivalence relation.

Am I missing something?

EDIT: I think the definition should say:

The equivalence relation $\sim$ is that which is generated by

$((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists d'', (f : d \to d''), (g: d' \to d'') with F(f)(x) = F(g)(x')) \,.$

This matches the definition of a direct limit on Wikipedia.