5 research outputs found
Elimination of quotients in various localisations of premodels into models
The contribution of this article is quadruple. It (1) unifies various schemes
of premodels/models including situations such as presheaves/sheaves,
sheaves/flabby sheaves, prespectra/-spectra, simplicial topological
spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors
in categories/strong stacks and, to some extent, functors from a limit sketch
to a model category versus the homotopical models for the limit sketch; (2)
provides a general construction from the premodels to the models; (3) proposes
technics that allows one to assess the nature of the universal properties
associated with this construction; (4) shows that the obtained localisation
admits a particular presentation, which organises the structural and relational
information into bundles of data. This presentation is obtained via a process
called an elimination of quotients and its aim is to facilitate the handling of
the relational information appearing in the construction of higher dimensional
objects such as weak -categories, weak -groupoids and
higher moduli stacks.Comment: The text is the same as in v6; this version contains corrections to
the published MDPI paper, the main reason for this change is that the diagram
of Proposition 3.1 was meant to be a 3 dimensional diagram (while only the
front face appeared in the published paper). The wording of some sentences
and the diagram of Example 6.42 are changed accordingly. A typo in the table
of Ex. 6.42 is correcte
Backprop as Functor: A compositional perspective on supervised learning
A supervised learning algorithm searches over a set of functions
parametrised by a space to find the best approximation to some ideal
function . It does this by taking examples , and updating the parameter according to some rule. We define a
category where these update rules may be composed, and show that gradient
descent---with respect to a fixed step size and an error function satisfying a
certain property---defines a monoidal functor from a category of parametrised
functions to this category of update rules. This provides a structural
perspective on backpropagation, as well as a broad generalisation of neural
networks.Comment: 13 pages + 4 page appendi