# Category Theory: Limits Clarification (Through Analogy)

I recently watched this video. After the tutor explained how to get terminal objects, products and pullbacks using limits it got me thinking about what exactly is going on and I came up with the following analogy. However, I'm new to category theory, so I'm unsure whether I'm on the right track.

To construct a limit you need a "shape" category, S, with two functors (one of which is a constant functor, c, the other , d, maps the diagram), and a category with objects of interest, O. From this, I figured that S is actually a "pattern" to be found, and d is a functor which finds all instances of the pattern. Element v is some object in O and element u is the "best" object (or rather, limit). As an analogy:

• S could be considered a regular expression (regex) pattern.
• d would be the matching algorithm.
• O is the body of text to search over.
• c maps to all letters in the text (assuming all letters are considered unique objects).
• v is the first letter of some text matched by the regex
• u is the first letter in the best match found by the regex (assuming the user selects the specific match they were after, and therefore all other matches point to it)

Given this analogy, when S is empty, it acts as a wild card, stating that everything is a possible match, and as such O becomes considered totally ordered, and the arrows point to the terminal object.

For cases when S is not empty, category O may be partially ordered, but the pattern matching + "link to all elements in the base of the cone" requirements limit the set of possible v candidates (the ones which fail to match are discarded and no longer considered). Then this sub-category is considered totally ordered, and the arrows point to the terminal object of this new category.

As a quick summary, is the S category a pattern to be matched which, when applied to O using d & c, creates a (hopefully) totally ordered sub-category of O? Hence, does this effectively reduce limits to being "terminal objects of totally ordered categories"?

• This doesn't make much sense. If you wanted to think of S as a pattern then d would be a single instance not an "algorithm", e.g., if S is a category with two objects and no non-identity arrows, then d is just two particular objects of O. There is no "total ordering" involved. See Goguen's A Categorical Manifesto for some CS-oriented analogies for categorical concepts and his "What is Unification?" In that context, a limit is a most general unifier for a system of equations represented by d. Jul 15, 2017 at 20:57
• @DerekElkins I shall have a look through A Catagorical Manifesto. Yes, my intuition does seem to be wrong as there would be a functor per "matched pattern" (and hence, multiple d's). Given this kind of context, then, what would u & v be? Is my analogy over those correct? If it is a pattern finder + sorter then I think I have a working idea of how it works. Thanks for responding so quickly =) Jul 16, 2017 at 8:36