Category Theory (CT) is  a branch of abstract algebra which is appropriate to formalize and relate different theories in mathematics and computer science. The level of abstraction of CT allows to discover new relationships between different theories, showing that frequently several  mathematical constructions are instances of a more general concept. Thus, CT is a powerful tool in conceptualizing and relating formal theories. Applications of CT to Philosophy (in particular, to Formal Ontology) are still incipient, although promissory.
From Lawvere's development of the theory of functorial semantics in 1963, CT shows that it is also an important tool in formal logic. Within this framework, a logic theory corresponds to a category, an interpretation is functor, a model is a functor to SET (the category of sets and functions), and a model homomorphism corresponds to a natural transformation.
The introduction of the concept of (elementary) topos by Lawvere and Tierney shows that CT has also an important role in Foundations of Mathematics. The concept of topos relates notions from topology, algebraic geometry and set theory, together with intuitionistic logic. A fixed topos can be seen as a given mathematical domain, where it is possible to develop concepts and constructions using its internal logic: (higher-order) intuitionistic type theory.
In recent years, CT has also shown to be a useful tool for representation of abstract logics, by defining appropriate categories of languages in which the logics are based. Using the representations, different process of combination of logics such as fibring can also be defined and studied as categorial constructions.
In this introductory tutorial we will give the basic notions of CT and we will show several applications to representation of abstract logics. We will start from the basic concepts from CT which permit to define the notion of topos.
Finally, we will study applications of CT in the representation of abstract logics, starting with a discussion about how can be defined categories of languages. The fundamental notion of morphism between logics, or translation between logics, will be analyzed in detail.

References:

Category Theory:
- R. Goldbaltt.  "Topoi: The categorial Analysis of Logic". North-Holland, 1984 (second edition).
- M. La Palme Reyes, G. Reyes and H. Zolfaghari. "Generic figures and
their glueings: A constructive approach to functor categories". Polimetrica, 2004.
- S. Mac Lane and I. Moerdjik. "Sheaves in Geometry and Logic". Springer, 1992.
Applications:
- A few papers about categorial fibring from the FibLog project .
- M.E. Coniglio. "Towards a stronger notion of translation between logics". Manuscrito 28, no. 2: 231-262, 2005.