# Trying to find the terminal object (and category) for this universal property

Consider an index set $$I = \{1, 2, \ldots, n\}$$. Let $$V$$ be a vector space and $$v_I$$ be a basis that is, $$\{v_1, v_2, \ldots, v_n\}$$. Let $$\phi: I \rightarrow V$$ be a map that maps the index set to a basis vector $$i \xrightarrow[]{\phi} v_i$$. Let $$W$$ be another $$n-$$dimensional vector space and $$f: I \rightarrow W$$ be any map. Then we have the following commutative diagram:

This looks like a universal property since: for all $$W$$ and $$f$$, there exists a unique linear transformation $$T: V \rightarrow W$$. This is true since $$T$$ is specified by where the basis elements $$v_i$$ go into $$W$$.

Now my question: if there is a universal property, there needs to be a certain category and a certain terminal object. What are these in this case?

It seems like the morphism $$\phi$$ (or really the pair $$(I,\phi)$$ is the initial object in a category where the objects are morphisms in the ambient diagram and morphisms are the commutative diagrams.

If the ambient category is $$\text{Vect}$$ where $$V,W$$ are objects but $$I$$ is not... I am confused.

This is indeed a universal property, and we can indeed find a category where $$(V, \phi)$$ is initial.

The category that you're looking for is a certain kind of comma category.

Here's a direct definition.

Objects are pairs $$(U, g)$$, where $$U$$ is a vector space and $$g$$ is an ordinary function $$g : I \to U$$.

An arrow from $$(U, g)$$ to $$(W, h)$$ is a linear map $$T : U \to W$$ wuch that $$h = T \circ g$$.

Composition is just ordinary composition of functions. It's easy to verify the axioms of a category from here.

Then we see that the claim "$$V$$ is the free vector space on $$I$$" is simply stating that $$(V, \phi)$$ is the initial object.

This is related to notions of the representability of a functor and of adjoints.

Edit: if you insist that a universal property be stated in terms of a terminal object, just take the opposite category of the one I have described, which makes $$(V, \phi)$$ into a terminal object.

• And if the OP insists that a universal property should always be interpreted as a terminal object in a certain category (this is a silly convention, but it seems to be implicit in the question), you can just consider the opposite of the category you described in your answer. Sep 28 at 20:08
• Thanks.. got it.. very helpful.. just for clarity I used terminal to mean either initial or final.. but was useful to help with the notion of the opposite category. Sep 28 at 20:56
• @MathDilettante Terminal object and final object are synonyms. If you use the word terminal to describe an initial object, you'll just confuse people. Sep 28 at 23:44
• @AlexKruckman I see that now.. thx Sep 29 at 10:40