Yoneda embedding target - MathOverflow

Yoneda embedding target

2010-01-01T18:43:07Z

I'm learning about representable functors from Vistoli notes thanks to Charles Siegel's answer.

I see that any category $\mathcal C$ can be embedded into $\text{Hom}\,(\mathcal C^{op}, \mathcal Set)$ by means of Yoneda embedding. I wonder if there are examples where the latter category would be interesting in itself, other then for these purposes?

Answer by Reid Barton for Yoneda embedding target

2010-01-01T18:50:54Z

Sure, take the category of simplicial sets, which is Hom(Δ^op, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

Answer by Mike Shulman for Yoneda embedding target

2010-01-01T18:55:45Z

Lots! Categories of that form (when C is small) are often called "presheaf categories". Many interesting categories are presheaf categories, such as simplicial sets, cubical sets, symmetric sets, etc. In particular, any presheaf category is a topos, and many interesting toposes are presheaf categories. The category of G-sets for any discrete group G is another nice example, since G can be regarded as a groupoid, hence as a category. Presheaves on a topological space are also interesting, if only as a means to the construction of sheaves. And simplicial presheaves on a category C (which are the same as presheaves on $C\times \Delta$) are sometimes easier to work with (once you put a nice model structure on them) than simplicial sheaves.

Many other interesting categories are full subcategories of some presheaf category; in fact a category is a full subcategory of a presheaf category as soon as it has a small dense subcategory. Thus, in particular, any accessible category is a subcategory of a presheaf category. This includes almost any "algebraic" category, such as groups, rings, fields, Lie algebras, etc.

Answer by Andrej Bauer for Yoneda embedding target

2010-01-02T00:16:43Z

Various categories of graphs are presheaf categories.

The category of directed graphs is (equivalent to) presheaves on $C$, where $C$ is a category with two objects, call them $V$ and $E$, and two parallel morphisms $s, t : V \to E$. If you have never seen this example, you should compute for yourself that a functor $G : C^{op} \to \text{Set}$ is the same thing as a directed graph. You may find this "Guided tour of the topos of graphs" illuminating.

Other categories of graphs are (almost) presheaf categories. For example, take the monoid $M$ of all endomaps $\lbrace 0,1\rbrace \to \lbrace0,1\rbrace$. This is a four-element monoid whose elements are the identity $id$, two constant maps $0$ and $1$, and the "twist" map $t$. View $M$ as a category (one object, four morphisms). The presheaves on $M$ are what is sometimes called "reflexive" graphs. Since this is not apparent at first sight, let me spell it out a bit. Consider a fuctor $F : M^{op} \to \text{Set}$, which is the same thing as a set $S$ with a right action of $M$. The corresponding graph $G$ has as its vertices the set $V = \lbrace x \in S \mid x \cdot 0 = x\rbrace$ of elements fixed by the action of the constant map $0$ (exercise: the points fixed by the constant map $0$ are the same as the points fixed by the constant map $1$). The edges of $G$ are the elements of $S$. An edge $e \in S$ has as its source the vertex $e \cdot 0$ and the target $e \cdot 1$. But since we also have the action of the twist map $t$, the situation is symmetric: to every edge $e$ going from $e \cdot 0$ to $e \cdot 1$ there corresponds the opposite edge $e \cdot t$ going from $(e \cdot t) \cdot 0 = e \cdot 1$ to $(e \cdot t) \cdot 1 = e \cdot 0$. So we are talking about symmetric graphs. Our graphs may be degenerate in the sense that an edge $e$ could be its own opposite (and then it is also a loop since $e \cdot 1 = e \cdot 0$). The graphs are reflexive because a homomorphism between them is allowed to "squish" edges to vertices, which is another exercise in computing natural transformations.

All of this and more (perhaps too much) can be found in:

Categories of spaces may not be generalized spaces as exemplified by directed graphs, F. William Lawvere, Revista Colombiana de Matematicas, XX (1986) 179-186. (Republished in: Reprints in Theory and Applications of Categories, No. 9 (2005) pp. 1-7)

Answer by Andrej Bauer for Yoneda embedding target

2010-01-02T00:27:39Z

Just thought of another one: Kripke models of intuitionistic logic are (very closely related to) presheaves on the poset of possible worlds. The interpretation of intuitionistic logic in a Kripke model coincides with the internal logic of the corresponding presheaf topos.