# Question about the concept of limits/colimits

Recently, I asked in a post, questions concerning the concept concerning concerning "limit for the diagram". For reasons probably due to my questions not clear or there were too many in that post, I will just repost here again being a bit more clear this time. For clarity and convenience purposes, the definitions and quoted texts are taken from $$\textit{Arrows, Structures and Functors the categorical imperative}$$ by Arbib and Manes

$$\quad\textbf{Definition 1}$$ A $$\textbf{diagram}$$ $$D$$ in a category $$\textbf{K}$$ is a directed graph whose vertices $$i$$ are labelled by objects $$D_i$$ of $$\textbf{K}$$ and whose edges $$i\rightarrow j$$ are labelled by morphisms $$\textbf{K}(D_i,D_j)$$

$$\quad$$A $$\textbf{cone}$$ for a diagram $$D$$ is a family $$X\rightarrow D_i$$ of morphisms from a single object $$X$$ such that $$X\rightarrow D_i\rightarrow D_j=X\rightarrow D_j$$ for every $$D_i\rightarrow D_j$$ in $$D$$. A morphism from a cone $$(X\rightarrow D_i)$$ to a cone $$(X'\rightarrow D_i)$$ is a $$\textbf{K}$$-morphism $$X\rightarrow X'$$ such that $$X\rightarrow X'\rightarrow D_i=X\rightarrow D_i$$ for all $$i$$. The cones for $$D$$ then form a category, and a $$\textbf{limit for the diagram}$$ $$D$$ is a $$\textit{terminal }$$ object in this category, i.e., $$(X\rightarrow D_i)$$ is such that for all cones $$(X'\rightarrow D_i)$$ on $$D$$ there is a $$\textit{unique}$$ morphism $$X\rightarrow X'$$ such that $$\color{red}{X'\rightarrow X\rightarrow D_i=X'\rightarrow D_i.}$$ We say that a limit $$(X\rightarrow D_i)$$ for $$D$$ has the $$\textbf{universal property}$$ with respect to cones $$(X'\rightarrow D_i)$$. $$\quad$$We say that a family $$(D_i\rightarrow X)$$ is a $$\textbf{colimit}$$ for $$D$$ if it is a limit for $$D$$ considered in $$\textbf{K}^{op}.$$ We say that a colimt has the $$\textbf{couniversal property}$$ with respect to $$\textbf{cocones}$$ $$(D_i\rightarrow X').$$

$$\textbf{Question:}$$

What I would like to know is: what is the definition for the "limit" for a diagram. I consulted a few texts in category theory, and all of them uses the notion of "functors". As Arbib and Manes' have not defined what a functor is at the point of the text where the "limit" notion has been introduced. I am wondering if anyone can explain to me what the concept of "limit/colimit" is without resorting to the concept of "functor". In the text, I see exercises like:

Use exercise 3.2 to conclude that the analogous construction to 19 does not construct colimits of diagrams in $$\textbf{Poset}.$$

or

"Prove that every small diagram (see exercise 2.4.13) in $$\textbf{Top}$$ has a limit.

I am having trouble with them since I don't know what the precise definition or a precise understanding of what a limit/colimit is. Also, is the notion of limit/colimit always referred to the concept of a diagram (commutative or not), etc?.

The notion of limit and colimit essentially always refers to diagrams of some sort. I am very surprised that functors are not yet introduced since the notion is straightforward and it allows for a more flexible idea of ‘diagram’. Almost always in my experience, diagrams are functors out of small categories and (co)limits are only considered for ‘small diagrams’. In principle the discussion can continue for diagrams out of large categories (i.e. directed graphs too big to be encoded as sets).

The limit (colimit) of a diagram are precisely defined as in the text (at least, that is one equivalent way of defining them). They are terminal (initial) objects in the category of cones (cocones) over (under) that diagram. This uniquely specifies up to isomorphism (in the cone (cocone) category), when they exist.

Limits and colimits can be thought of as the object whose data most closely represents the diagram (from above or from below). For any other attempt to amalgamate the diagram into one place (this is my informal description of a (co)cone) must (uniquely) factor through the (co)limit object.

• so is the definition for a cone always refer to a triangle shaped commutative diagram? Also, what did you mean by under/over?
– Seth
Commented Feb 15, 2023 at 11:48
• @Seth Forget commutative diagram (the paths in a diagram don’t have to commute). We just expect that, if $a,b$ are objects in the diagram and $x,y:X\to a,b$ are the cone morphisms (the legs) then $y=f x$ for every possible $f:a\to b$ in the diagram. “Under/over” is just the terminology I’ve seen for this, it is purely conceptual language. Cones go ‘over’ a diagram and cocones go ‘ynder’ Commented Feb 15, 2023 at 11:50
• kk. So what does it mean to say that it is/is not possible to construct a limit/colimit for a particular category?
– Seth
Commented Feb 15, 2023 at 11:52
• @Seth Categories don’t necessarily have all limits or all colimits. It’s nice when they do… to say we ‘construct’ a limit is again just conceptual language. This means a standard ‘process’ or more accurately a standard description of the limit object so that: (a) you know it exists and (b) you have a good idea of what it ‘looks like’ (eg in the category of sets I know exactly what the elements are for the limit of a diagram). Commented Feb 15, 2023 at 11:56
• The exercise about Poset is just saying, whatever the construction they’re talking about is, it will produce something, but that something won’t be a colimit. Commented Feb 15, 2023 at 11:57