Span categories provide an abstract framework for formalizing mathematical models
of certain systems. The mathematical descriptions of some systems, such as classical
mechanical systems, require categories that do not have pullbacks, and this
limits the utility of span categories as a formal framework. Given categories
and
and a
functor
from
to
, we introduce the notion
of an
-pullback of a cospan
in
, as well as the notion
of span tightness of
.
If
is span tight, then we can form a generalized span category
and circumvent the technical
difficulty of
failing to have
pullbacks. Composition in
uses
-pullbacks
rather than pullbacks and in this way differs from the category
, but reduces to
it when both
has
pullbacks and
is the identity functor.
