# Spans in Manifold

Famously, the category of manifolds (which I will call \mathsf{Man}) does not have all pullbacks. But you can take pullbacks against submersions, and also submersions are stable under pullback and composition.

So is there a (double) category where the objects are manifolds, and the morphisms (proarrows) are spans where, say, the right hand leg is a submersion? It seems like this works, but I’m not confident in my differential geometry. Assuming that this does work, call this \mathsf{ManSpan}.

This came up while investigating the following construction. Let Q be a set, and let \mathrm{Spc} \colon Q \to \mathsf{Man} be a function. Then let \mathsf{C} be the category of cospans I_1 \to J \leftarrow I_2 in \mathsf{FinSet}/Q where the left hand map is epic and the right hand is monic. These kind of cospans were introduced to me by @david.jaz, but I don’t have a resource for them or know what the normal name for them is.

There is then a functor \mathrm{Spc}^\ast from (\mathsf{FinSet}/Q)^\mathrm{op} to \mathsf{Man} defined by sending an object (I, \tau \colon I \to Q) to the space \prod_{i \in I} \mathrm{Spc}(\tau(i)), and sending a morphism a \colon I \to J to the precomposition function f = a^\ast \colon \mathrm{Spc}^\ast(J) \to \mathrm{Spc}^\ast(I) defined by f(y)_k = y_{a(k)} for k \in I.

If a is monic, then a^\ast is a submersion. Thus I believe that \mathrm{Spc}^\ast extends to a functor \mathrm{Spc}^+ from \mathsf{C} to \mathsf{ManSpan}.

But the image of \mathrm{Spc}^+ consists of spans with even nicer properties: the left leg is an embedding and the right leg is a surjective submersion! This makes me wonder, is my life going to be easier or harder if I define \mathsf{ManSpan} to require this; i.e. I just consider spans where the left leg is an embedding and the right leg is a surjective submersion.

Any comments or references on this sort of thing are welcome.

This doesn’t answer your question but I can’t help myself. I know there’s a general adversion in the wider literature to using simpler and weaker notions of smooth space than “manifold”, but may I recommend using diffeological spaces or more general smooth sets (sheaves on euclidean spaces (or open subsets of euclidean spaces) equipped with the open covering topology). These have all pullbacks, and the evident embedding which interprets a manifold as a diffeological space or smooth set preserves transverse pullbacks — so you don’t lose out on anything. They also include Banach and Frechet manifolds fully faithfully, so that you can seamlessly go to infinite dimensional spaces (without faking it 'till you make it, which seems to be the usual approach). On the other hand, diffeological spaces are a quasitopos and smooth sets are a full topos, meaning that they have really nice structure that smooths out whatever you’re trying to do. Pre-symplectic structures are even representable in the topos of smooth sets! This means that you can slice over the representing object and get a topos which fully faithfully includes symplectic manifolds and symplectomorphisms, and whose spans include symplectic or Lagrangian relations. See the relevant section of Urs’ Geometry of Physics.

You don’t even have to abandon manifolds — which of course you won’t want to, since all the theorems are proven for them and it’s not always clear which ones will generalize. Rather, work at this greater level of generality for your theory, and then check that the various constructions you do in particular preserve the property of being a manifold. This is effectively what you’re already doing — it’s no extra work — but has the nice side effect of making it very clear how to generalize from manifolds to infinite dimensional spaces and function spaces.

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As a second reply, let me advertise the double category of cospans of finite sets whose left leg is surjective and whose right leg is injective.

First, conceptually these are easier to understand as a model of variable sharing: the domain is our current set of variables (or variable names) and the codomain the new set of variable names. The left leg is a partition of the current set of variables; we will identify all the variables in a block of this partition. The right leg selects, for each new variable name, which block of current variables it will refer to. That is, such a cospan represents a two-step process in variable sharing: set variables equal, and then re-expose (or, dually, hide). The category can be axiomatized as the PROP with a binary operation e : 2 \to 1 and a nullary co-operation h : 1 \to 0 subject to the axiom that e is associative. These correspond respectively to the cospans 2 \to 1 = 1 and 1 = 1 \hookleftarrow \emptyset.

Graphically, the condition that the left leg be surjective and the right leg be injective means that every external port is live (connects to some internal port) and there are no passing wires (no two external ports are connected).

Theoretically, this double category is nice. It is thin by the surjectivity of the left legs: a morphism of diagrams is determined by its action on ports, with its action on internal nodes deducible. It is also both “spanish” and “cospanish”, by which I mean that the Pare’ representable functors of either variance are pseudo. This follows from the adhesivity of the category of finite sets, which implies in particular that the pushout square defining a composite is also a pullback; this property uses the fact that the right leg is injective.

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This question came out of a write up for a paper that will appear in a very applied journal, so I don’t think that diffeological spaces are going to fly… But in any case you can interpret my question as asking what conditions will keep me in the full subcategory of manifolds!

You gave me neither an answer to this, nor a good name for cospans with monic and epic legs!! I already liked them I just wanted to know what they were called!!

Haha, “smooths out what you’re trying to do”, I see what you did there

The epic-monic cospans are a way of presenting subquotients, which are a big deal in abelian-land. That’s all I can think of on that front.

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I call them “rhizomes” on account of what the diagrams look like (little potatoes overgrowing).

But I never wrote anything on it, so it’s your choice in what to name them.

Kevin is right that they are also “subquotients”.

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Rhizome is an awesome name, especially for confusing art people.

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