Discrete category

In mathematics, in the field of , a discrete category is a category whose only morphisms are the identity morphisms: homC(X, X) = {idX} for all objects X homC(X, Y) = ∅ for all objects X ≠ Y Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set | homC(X, Y) | is 1 when X = Y and 0 when X is not equal to Y. Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category. Any class of objects defines a discrete category when augmented with identity maps. Any of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are . The of any functor from a discrete category into another category is called a , while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be used as a or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category C and the discrete category 2, one can consider the C2. The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product. The functor from Set to Cat that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects. (For the right adjoint, see .

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Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

Andreas Loukas, Nikolaos Karalias

Integrating functions on discrete domains into neural networks is key to developing their capability to reason about discrete objects. But, discrete domains are (I) not naturally amenable to gradient-based optimization, and (II) incompatible with deep lear ...

2022

A multiphase CMAQ version 5.0 adjoint

Athanasios Nenes

We present the development of a multiphase adjoint for the Community Multiscale Air Quality (CMAQ) model, a widely used chemical transport model. The adjoint model provides location- and time-specific gradients that can be used in various applications such ...

2020

ANISORROPIA: The adjoint of the aerosol thermodynamic model ISORROPIA

Athanasios Nenes

We present the development of ANISORROPIA, the discrete adjoint of the ISORROPIA thermodynamic equilibrium model that treats the Na +-SO 4 2- HSO 4 -NH 4 +-NO 3 -Cl -H 2O aerosol system, and we demonstrate its sensitivity analysis capabilities. ANISORROPIA ...

2012

Diagonal functor

In , a branch of mathematics, the diagonal functor is given by , which maps as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the : a product is a universal arrow from to . The arrow comprises the projection maps. More generally, given a , one may construct the , the objects of which are called . For each object in , there is a constant diagram that maps every object in to and every morphism in to .

Diagram (category theory)

In , a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.

Pullback (category theory)

In , a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the of a consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written P = X ×f, Z, g Y. Usually the morphisms f and g are omitted from the notation, and then the pullback is written P = X ×Z Y. The pullback comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms.