Abstract Nonsense

Crushing one theorem at a time

Universal Arrows and Universal Elements (Pt. I)


Point of Post: In this post we discuss the notion of universal arrows and universal elements, this is in preparation for our discussion of particular universal problems such as limits and colimits.

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Motivation

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Category theory has the ability to bring together disparate ideas in mathematics under one blanket theory. Category theory accomplishes this in two ways. Firstly, it provides a universal vocabulary which allows us to have a coherent discussion. Secondly, it provides a way of thinking, a way of formulating questions that brings forth the truly important characteristics of an object, separating it from the confounding chaff. In this post we describe a concept that epitomizes both of these categorical methodologies–universal characterizations.  The idea behind universal characterizations are simple: in a given category what we really care about are arrows, and thus we should (when possible) define an object entirely in terms of arrows. To be more explicit it would be preferable to define a construction (say) in a way such as “this construction is the unique, up to isomorphism, object such that given maps out of…” opposed to “this construction is defined to be this object, and any object isomorphic to it”. It is clear that defining a construction in the first way, gives a more substantive description about what the construction is “all about”. For explicit example, consider that in the category of abelian groups we could define the “product” of two abelian groups A,B to be either “an abelian group P along with group homomorphisms A\overset{p_1}{\longleftarrow}P\overset{p_2}\longrightarrow B such that given an abelian group C and group homomorphism f:C\to A and g:C\to B there exists a unique group homomorphism j:C\to P such that p_1\circ j=f and p_2\circ j=g” or we could define it as “any group isomorphic to A\times B (defined in the usual way)”. A little experimentation shows that these two description are describing the same isomorphism class of objects in \mathbf{Ab}, but it’s clear that the first definition clarifies the “use” of the product of groups–what qualities “make it a product”. To drill it in, a definition by mappings tells us how an object is defined, by telling us how it relates (by arrows) to other objects in the category. The idea of defining objects in a way involving only mappings exemplifies the second method of category theory, the rest of this post shall be devoted to the first–the development of the proper language that will allow us to precisely phrase these “universal mapping problems”.

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Universal Arrows and Universal Elements

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So, as stated we’d like to define the general scenario in which we can define objects as solutions to certain “universal mapping problems”. The first step in doing this is to define what we mean by “universal”. To this end, suppose that we have two categories \mathcal{C} and \mathcal{D} and a functor F:\mathcal{C}\to\mathcal{D}. We then define for an object d\in\mathcal{D} a universal arrow from d to F to be an ordered pair (c,u) where c is an object in \mathcal{C} and u is an arrow d\to F(c) satisfying the following property: given any other object x in \mathcal{C} and an arrow d\xrightarrow{f}F(x) there exists a unique arrow c\xrightarrow{j}x such that F(j)\circ u=f.

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There’s a nice pictoral way to think about universal arrows, but unfortunately my only means of drawing it is by a crude online sketch pad, regardless here’s what the picture looks like

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Thus, we can see that a universal arrow is basically a \mathcal{C}-object c and an arrow between the image under F of this object and d such that any time there is another \mathcal{C}-object x and an arrow from d to x‘s image under F then we can find a unique arrow c\to x whose image under F commutatively closes up the triangle started by F(x)\longrightarrow d\longleftarrow F(c).

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There is another way to interpret universal arrows which shows their uniqueness. Indeed, we have the following theorem:

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Theorem: Let \mathcal{C} and \mathcal{D} be two categories and F a functor \mathcal{C}\to\mathcal{D}. Then, if d is an object of d an ordered pair (c,u) where c is an object in \mathcal{C} and u an arrow d\to F(c), is a universal arrow from d to F, if and only if (d,c,u) is an initial object in the comma category (d\downarrow F).

Proof: Suppose first that (c,u) is a universal arrow from d to F. We note then that for any other object (d,x,f) in (d\downarrow F) the fact that f is an arrow d\to F(x) guarantees us a unique arrow c\xrightarrow{j}x such that F(j)\circ u=f. We then note that (1_d,j) is an actual arrow in (d\downarrow F) from (d,c,u) to (d,x,f), and moreover if there were another arrow (1_d,k) then k would need to be an arrow c\to x satisfying F(k)\circ u=f, contradicting the uniqueness of the mapping j. Thus, we see that every object in (d\downarrow F) has a unique arrow emanating from (d,c,u) and so (d,c,u) is initial in (d\downarrow F).

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Conversely, suppose that (d,c,u) is initial in (d\downarrow F). Then, if x is an object in \mathcal{C} and f is an arrow d\to F(x) then (d,x,f) is an object in (d\downarrow F). Since (d,c,u) is initial we know there exists an arrow (1_d,j) in (d\downarrow F) from (d,c,u) to (d,x,f). Thus, we see that j is an arrow c\to x such that F(j)\circ u=1_d\circ f=f. Moreover, it’s clear that such an arrow is unique, otherwise it would create two arrows from (d,c,u) to (d,x,f) in (d\downarrow F) contradicting the definition of an initial object. \blacksquare

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From first principles we can conclude from this that if (c,u) and (c',u') are both universal arrows from d to F then there exists a unique isomorphism c\xrightarrow{t}c' such that F(t)\circ u=u'.

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Ok, so now that we have a conceptual understanding of what universal arrows are, let’s take a look at some typifying examples.

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Let R be some unital ring, consider the forgetful functor U:R\text{-}\mathbf{Mod}\to\mathbf{Set}. Choose some object (set) X in \mathbf{Set}. We claim then that the free left R-module R[X] and the set map i:X\hookrightarrow R[X] is a universal arrow from X to U. Indeed, given any left R-module M and any set map f:X\to M we know there exists a unique R-map \widetilde{f}:R[X]\to M such that U(\widetilde{f})\circ i=f, which is precisely the definition of a universal arrow from X to U.

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For another example of a universal arrow to a forgetful functor, consider the forgetful functor U:\mathbf{Top}\to\mathbf{Set}. Choose some set X, we claim that an example of a universal arrow from X to U is the discrete space X_d (i.e. just the discrete topological space, with underlying set X–we only used d to differentiate between X and X_d) and the identity set map \text{id}_X:X\to X=U(X_d). Indeed, let Y be some other topological space and suppose that we have a set map f:X\to Y. Of course, since the map f:X_d\to Y is also continuous since X_d is discrete (all maps out of it are continuous!) and satisfies f\circ\text{id}_X=f. Moreover, it’s fairly evident that this is the only such map.

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Remark: In general, and it’s not hard to see this if one writes down the mapping properties of such an object, universal arrows to forgetful functors are “free” constructions in some sense.

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. Introduction to Homological Algebra. Springer-Verlag. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.

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January 12, 2012 - Posted by | Algebra, Category Theory | , , , , , ,

6 Comments »

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