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I was wondering whether the following category already has been used somewhere and whether it already has been named.

Let us fix a field $k$ (or more generally a ring). An object is just a $k$-vector space and a morphism $f:U\rightarrow V$ consists of a choice of a sub-vectorspace $V_f\subset V$ and a well-defined linear map $\overline{f}:U\rightarrow V/V_f$.

The composition of $f:U\rightarrow V$ and $V\rightarrow W$ is given by $$W_{g \circ f} = p_{W\rightarrow W/W_g}^{-1}(g(V_f)),$$ $$\overline{g\circ f}(x) := [\overline{g}(y)],$$ where $p_{W\rightarrow W/W_g}$ denotes the canonical projection $y\in V$ is a representative of $\overline{f}(x)\in V/V_f$.

The reason behind this is that I wanted to compose maps this way quite often, and one could either

  • give a new name to $V/V_f\rightarrow (U/W_g)/g(V_f)$ and work in the category of vectorspaces. Now I would have to introduce a lot of new names for the map $g$ depending on with which maps I compose them, which makes things worse to read.

  • Try to lift each map $U\rightarrow V\rightarrow V/V_f$ to a map $U\rightarrow V$. This involves a choice which might affect the rest of the arguments. Now one has to argue that things do not depend on the choice for no other reason than that we have introduced

  • The naive idea that there should be a functor from this category back to vectorspaces sending $f$ to $\overline{f}$ does not work, since this does not respect composability (The target of $f$ is $V/V_f$ and the source of $g$ is $V$.

So I think using this category might be the cleanest way and so I am wondering whether it has already been used/named before.

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    $\begingroup$ This seems to be a subcategory of the category of vector spaces and correspondences, where the correspondences are required to be defined on the whole source (a correspondence $U\to V$ is just a subspace of $U\oplus V$, your "not well-defined morphisms" embed by taking their graph). $\endgroup$
    – Denis Nardin
    Commented Apr 9, 2020 at 12:00
  • $\begingroup$ Thank you. Do you want to post it as an answer ? Maybe the name 'total linear correspondence' makes sense? edit: the linearity contraints are already dealt with in your comment by taking a subspace of $U\oplus V$ and not an arbitrary subset. $\endgroup$
    – HenrikRüping
    Commented Apr 9, 2020 at 12:20
  • $\begingroup$ I'll see if I can find the time to make it into an answer. But note that under my definition all correspondences are linear (each nonempty fiber over a point of $U$ is indeed an affine subspace parallel to the fiber over 0, etc.). I'd go with "total correspondences", perhaps $\endgroup$
    – Denis Nardin
    Commented Apr 9, 2020 at 12:23
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    $\begingroup$ I'd suggest "linear relations" as an alternative $\endgroup$
    – Neil Strickland
    Commented Jun 16, 2020 at 8:10
  • $\begingroup$ Linear relations (under that name, although his are not necessarily total) are used in MacLane's book on homology to write down, for example, connecting maps in long exact sequences arising from short exact sequences of complexes. It allows him to avoid the phrase «let us check that the function $f$ is well-defined». $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 23, 2022 at 22:37

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As Denis Nardin pointed out in the comments, the name total correspondence is a good choice.

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