      a       ISOMORPHISM      x
     / \   <··············>   / \
    b   c                    y   z

      ^                        ^
      : HOMOMORPHISM           : HOMOM.
      :                        :

 b1---a---c3     MODEL      x1  x2
     /|\      <········>   /  \/  \
    / | \                 /   /\   \
  b2  c1 c2             y1  y2  z1  z2

   SYSTEM1                 SYSTEM2

Secondarily, there are the models M of M0-objects X.

A model is «[a] representation of some phenomenon of the real world made in order to facilitate an understanding of its workings. A model is a simplified and generalized version of real events, from which the incidental detail, or 'noise' has been removed» [x].

«A model can be understood as an image, however with the speciality that in it the original is represented only in part, while on the other hand it goes beyond the original. ... Model = a system that is related to another one in a way such that a homomorphism of one system is isomorphic to an homomorphism of the other one» [SuB 48, my translation].
Where 'homomorphism' ('isomorphism') is a mapping of the system's elements (which are related with each other) into an intermediate domain of related elements in a way such that elements related (and unrelated, for isomorphisms) in the system are mapping to elements that are also related (and unrelated, respectively).

An abstract model of concrete objects may be called a conceptualization. A scientific theory is such a conceptualization, i.e., an abstract model of some aspect of reality. The abstract model is obtained from the (sub)objects and the (sub)processes of an object-cum-operations by the process of abstraction, namely by aggregation abstraction, classification abstraction, and generalization abstraction. For example, a UML model of the Golden Gate might consist of Steelbar objects, Nut objects, Bolt objects, Steelcable objects, etc., connected by screwing links (between Nut and Bolt objects), interlocking links (between Steelbar(s) and Nut-and-Bolt-composites), and whatever connects the Steelcables with the Steelbars.

Representation. Being objects, non-physical models may require representation: Symbolic and mental models require some physical representation, like ink on paper or electric potentials in neurons. An abstract object M used as model for X presupposes a mental representation M' in the one who chose to use M as X's model. For communication, one can also produce a representation by a physical object, like a plastic representation M' of a coalesced-spheres model M of molecule X; or by a symbolic object (model description), like a UML-diagram M' representing a UML-model M of system X.

A system description, whether in textual or graphical form, is a linguistic artifact M (symbolic object) that is (a) symbolic model of a system X (the object) at M0 or (b) the symbolic representation, or description, of an abstract model M' of X. E.g., the specs of the Golden Gate bridge; or the UML-diagram of the UML-model of the Golden Gate (NB it would seem wrong to say that you could have a UML-diagram of the bridge itself). Besides system description by mathematical formulae, a range of special purpose modeling languages, "system description languages," have been developed in different fields of (hardware and software) engineering (-> the use of specialized languages). Computer programming can be seen as developing the description of a desired system (program), and giving it to the computer, which interprets the description and simulates the system described therein; that is, the programmed computer is a physical model of the desired system.

«A system is a part of the world which we choose to regard as a whole, separated from the rest of the world during some period of consideration, a whole which we choose to consider as containing a collection of components, each characterized by a selected set of associated data items and patterns, and by actions which may involve itself and other components.
We make mental and manifest models of a considered system. A system description is expressed in a system description language. A model is generated from a system description.»
[From the Delta report on the 1975 Delta project: Erik Holbæk-Hanssen, Petter Håndlykken, Kristen Nygaard: System Description and the Delta Language; Norwegian Computing Center 1977. scan of the relevant pages (doc)] ]

The Meta-Model / The Ontology / Grammar of the Modeling Language (Meta-Level M2)

Thirdly, there is the "language" of entities, relationships, processes, etc. from which an M1-model M (modeling some X) is built, the modeling language (e.g., the UML = Unified Modeling Language). (Strictly speaking, the term "language" would apply only to symbolic M1-models.) For example, the paper model of the Golden Gate might be made from Folded-paper-strips, Dot-of-glues, Glue-cohesion-bonds, etc. And the UML model would be made of Objects, Links, Classes, and Associations. The grammar of this language are the categories of the entities, relationships, processes, etc. in the model(s), and their combination rules. A particular conceptualization (abstract model) of this grammar is called a meta-model MM of M1-model M. It contains the class abstractions of the elements in M.

From the language of the model M, we have to distinguish the language of the symbolic representation M' of an (abstract) model M: The language of UML-models is not the same as the language of UML-diagrams.

As a class model, not instance model, of M, meta-model MM is not at a model of X: The Golden Gate bridge was not made from paper, glue, and cohesion-bonds, nor from UML objects, links, classes and associations, but, I assume, from steal bars, nuts & bolts, interlocking relationships, and so on.

As a class model, a meta-model MM obviously can have several instances, i.e., a whole family of M1-models can have the same meta-model. A meta-model can therefore be used to describe not just one particular model, but also more generally to define (specify) an entire category of M1-models of the same modeling language, i.e., satisfying a common grammar.

Ontology (or meta-physics, a branch of philosophy[x] with a long tradition [x]) can be understood as the persuit of the meta-model in terms of which any model of our particular world (the reality) would have to me expressed. It offers the world-modeling categories of things/objects/individuals, properties/attributes, relations, events, space, time, etc. Cf. history. Cf. Theory of Abstract Objects.

Representation. Being abstract models, all meta-models require representation. Being class models, there can be no direct physical representation (which would need to contain, e.g., all paper strips and glue dots). That is, meta-models have mental or symbolic representations (which in turn have some physical representation - but these are not transitively also representations of MM).

Being a model (namely of M - or of L?), meta-model MM is itself also built from a language of entities (namely classes, since it is a class model) and possible combinatorial relationships. The meta-model's language could be called the meta-modeling language LL (in analogy to calling the model's language a "modeling language"), or meta-language for short. The conceptualization of its grammer is at the meta-meta level M3 (not shown)!

However, one might also call "meta-language" the language of the meta-model's symbolic representation, e.g., UML-diagrams, which is not exactly the same as the meta-model's language, the meta-modeling language.

-> meta-semiotics