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Why Identity, Why Category Theory?
We are living in a moment of conceptual fatigue, if not the total decline of identity politics. Once a disruptive analytic, a lexicon for articulating marginalization and reclaiming visibility, in the last couple of decades this political strategy became a kind of ambient institutional noise. The terms affiliated with identity politics that once unsettled power became embedded in it and also used selectively as weapons of exclusion and cancellation. Categories like race, gender, sexuality, and disability appeared everywhere: in grant applications, museum wall texts, university syllabi, social media bios, and corporate advertisements. Yet their ubiquity failed to sharpen their political efficacy. Instead, identity calcified into procedural optics, a bureaucratic protocol of recognition and exclusion that often displaced the antagonisms it once named and covered it with a thick coat of paint.
At its core, the dominant discourse on identity has been built upon a model of classification. It assumes that people can be sorted, that attributes can be layered, and that justice can be algorithmically approximated through representational balance.
The 2016 and 2024 U.S. elections revealed the limits of identity politics as an electoral strategy. Hillary Clinton’s campaign focused on the symbolism of the first female president, while 2024 emphasized the historic candidacy of the first African American woman. In both cases, identity was treated as a moral achievement rather than a response to a systemic crisis. This approach alienated many voters facing economic and institutional disillusionment and gave the far right space to weaponize identity as a grievance. By relying on representation over transformation, the Democrats helped solidify the cultural terrain on which right-wing populism could grow.
This is how an unqualified and brutal prosecutor like Kamala Harris was selected by Joe Biden as his running mate in the midst of the BLM movement to appease black people. Even Intersectionality, though originally conceived as a critique of one-dimensional analysis, was operationalized as an additive matrix of social variables. Look at me, a Muslim gay Immigrant. My Intersectionality still might give me three credit cards, allowing me to say things and do things that my white male colleagues don’t dare to. This framework, despite its sophistication, is still rooted in a metaphysics of being: who are you? Which boxes apply? How many oppression points have you accumulated?
What we propose instead is a shift toward a politics of transformation. Rather than asking what identity is or to what group one belongs, we ask what identity does, how it moves, and how it composes and decomposes across social, political, and economic systems. To make this shift, we need a new language, one that privileges transformation over substance, relation over essence, process over position, and mapping over membership.
Category theory, developed in the 1940s by Samuel Eilenberg and Saunders Mac Lane, offers such a language.[1] Born within the realm of algebraic topology, it provides a general theory of mathematical structure grounded not in the properties of individual elements but in the morphisms and the structured relations between them. Category theory abstracts the notion of function beyond numbers and sets, formalizing how structures relate, translate, and preserve coherence across different contexts. It is not a theory of what things are but how they transform.
A way to grasp category theory through everyday life is by thinking about translating recipes across cultures. For example, adapting lasagna for a Japanese kitchen might involve using tofu cheese instead of mozzarella or rice noodles in place of pasta. What’s preserved isn’t the exact ingredients but the structure and relationships within the dish: its layering, balance, and richness. This mirrors what category theory calls a functor: a transformation that maintains coherence across different systems.
This paper introduces category theory not as a mathematical detour but as a conceptual tool to reframe how we think about identity under conditions of institutional capture, technological governance, and global reordering. If identity today is in crisis, it is not simply because it has been overused but because it has been both consciously and unconsciously misframed. Category theory offers the possibility of reframing identity not as an essence to be located but as a structure to be traced.
Our journey toward applying category theory to questions of identity was inspired by the groundbreaking work of Fernando Zalamea, particularly his Synthetic Philosophy of Contemporary Mathematics.[2] Zalamea’s exploration of Grothendieck’s mathematics as a philosophical system reveals how category theory offers not just technical tools but a profound conceptual framework for rethinking relation, difference, and transformation across domains. His emphasis on the dynamic transit between local and global perspectives in Grothendieck’s work suggested a powerful model for approaching the contextual nature of identity formation in our current cultural and political landscape.
Think of identity as a subway map rather than a fixed address. Who we are depends not on isolated traits but on how those traits connect and shift across different contexts: work, home, online, public, private. Category theory helps us see identity not as a static object but as a network of transformations between roles, environments, and social positions.
Historical Foundations of Category Theory
Category theory emerged during a pivotal period in the mid-20th century, as mathematics was undergoing a foundational transformation. Conceived by Samuel Eilenberg and Saunders Mac Lane in 1945 in their work on algebraic topology, category theory was not merely an abstraction but a response to a very practical problem: how to translate results across different mathematical domains with structural consistency. What began as a formalism to compare and relate different homology theories quickly matured into a sweeping reformulation of how mathematics could be practiced.
In category theory, an object is defined not by its internal content but by its relationships (its morphisms) to other objects. For example, consider two sets:
A={1,2,3} and B={x,y,z}.
From a category-theoretic perspective, we are not concerned with the specific elements of these sets but with how they are related through functions, such as a morphism f:A→B. If there exists a bijection between them, then A and B are considered structurally identical or isomorphic, regardless of their internal elements. This illustrates a core idea in category theory: identity is determined relationally through the network of morphisms rather than through internal characteristics.
At its core, category theory replaced the obsession with objects and their internal properties with a focus on morphisms (maps or arrows) between objects. It shifted attention from what things are to how they relate, how they transform, and how they preserve structure across those transformations. This emphasis on relation over substance was revolutionary. It allowed mathematicians to build vast unifying frameworks in which seemingly unrelated concepts (sets, spaces, functions, groups) could be described in a common language of arrows and compositions.
Building upon this foundation, Alexander Grothendieck expanded category theory into a powerful engine for reformulating algebraic geometry. Grothendieck was one of the most influential mathematical minds of the 20th century. He was born in Berlin in 1928 to anarchist parents, and his professional journey took him from statelessness to the prestigious Institut des Hautes Études Scientifiques (IHÉS) before his eventual withdrawal from the mathematical establishment. Through innovations like schemes, topoi, and sheaf theory, Grothendieck used categorical reasoning to describe spaces not in terms of their points but in terms of their relations to other spaces. He imagined a mathematics of gluing local data assembled into global coherence.
Schemes
Imagine a museum where each room lights the same Goya print differently (UV, candlelight, LED), revealing unique aspects. A scheme works similarly: it stitches together partial views, each shaped by local conditions, into a coherent whole. It’s a way to model complex spaces through overlapping, context-sensitive perspectives.
Topoi
Think of Netflix versus a public library. Both manage information, but each follows a different logic: algorithmic suggestions vs. neutral cataloging. A topos is like a self-contained world where such rules define what counts as meaningful or true. It helps us compare distinct, internally consistent systems of knowledge.
Sheaf Theory
Wikipedia has grown from many users editing locally; no one sees the full picture, but their contributions are assembled into coherent articles. Sheaf theory formalizes this: local data is patched into the global structure. In art, it’s like a community mural where sections are painted separately but designed to form a unified image.
In his Esquisse d’un Programme,[3] Grothendieck outlined a philosophical vision of mathematics as a landscape of transformations and correspondences. The objects of knowledge were not fixed entities but dynamic configurations, flexible to context and scalable across different levels of abstraction. As Jean-Pierre Serre noted in summarizing Grothendieck’s approach, “the objects of a category do not play a large role; it is the morphisms that are essential.”[4] This ethos resonates far beyond mathematics: it speaks to an epistemology in which systems are understood through flows, connections, and processes rather than fixed categories.
In this way, category theory is not just a technical development but a methodological shift with philosophical implications. It aligns with a long arc of relational thinking, from Spinoza’s modes and Leibniz’s monads to Whitehead’s process philosophy and Deleuze’s difference and repetition. It is a language that foregrounds transformation, and it is precisely this feature that makes it so powerful for rethinking identity today.
Set Theory vs. Category Theory: A Paradigm Shift
The dominant framework underpinning contemporary discussions of identity is still set-theoretic. From bureaucratic databases to academic theories of Intersectionality, identity is often conceived in terms of membership in predefined sets: a person is Black, queer, or disabled. These identities are seen as discrete units that intersect and accumulate. This logic carries the assumptions of classical set theory: objects belong or do not belong, categories are static, and operations are performed upon well-defined elements. However, this model has significant limitations. It flattens identity into combinatorial logic, ignoring the contextual transformations that occur as identities move across institutions, borders, or discourses. The set-theoretic view presumes a universal coordinate space in which identities are mapped, compared, and ranked as if the world itself were merely an elaborate spreadsheet.
The dominant, set-theoretic view of identity casts Barack Obama as Black, the first African American president. This is treated as a fixed, symbolic category of progress. This framing, however, obscures the relational roles he played: expanding drone warfare in Afghanistan and beyond, defending neoliberal economics against democratic socialism within his own party, twice intervening to defeat Bernie Sanders, and reinforcing U.S. military hegemony. His administration supported regime change in Ukraine by dispatching Victoria Nuland to back the Maidan uprising, escalating tensions with Russia and expanding NATO’s influence, eventually leading to Russian Aggression against Ukraine. In Libya, his interventionist policies helped topple Gaddafi, plunging the country into chaos and reintroducing the open-air slave trade. This outcome tragically contradicted his racial symbolism. Category theory shifts the question from “Who is Obama?” to “What does he enact within multiple systems?” It reveals how Obama’s Black identity functioned politically by masking his ill intentions and decisions. It was a morphism that translated symbolic progress into legitimacy, masking his role as a neoliberal conflict enabler. Identity here is not essence but a relational function operating across structures of global power.
Even when Intersectionality is deployed, its operationalization tends to revert to additive inclusion rather than transformative relations. If Obama’s racial identity helped legitimize neoliberal militarism, Kamala Harris’s elevation as the first Black and South Asian woman vice president similarly illustrates how intersectional identity can mask, rather than challenge, entrenched power. While Harris’s rise was framed as progress, her record as California’s attorney general tells a different story: she prosecuted Black Californians harshly for nonviolent drug offenses and resisted sentencing reforms. Internationally, she has offered unquestioning support for Israel’s war on Gaza, echoing the same imperial consensus as her Republican or White male counterparts. Category theory reveals how identity can act like a conduit through which empire and capital cloak themselves in symbolic diversity.
Category theory, by contrast, offers a way out of this impasse. It replaces the logic of classification with one of translation. Here, identity is no longer a fixed label or intersection of labels but a morphism. It is a structured transformation from one domain to another. The emphasis shifts from belonging to behavior, from static coordinates to dynamic mappings.
In categorical terms, we are less interested in which set a person belongs to and more in how their position transforms as they traverse different systems: legal, social, medical, and artistic. Identity becomes not a point but a path.
This shift resonates with lived experience: how one’s gender, for instance, is legible in one context but opaque in another; how race operates differently in Europe than in North America; how queerness is entangled with language, geography, and temporality.
Consider a wealthy, closeted gay man from Saudi Arabia with a heterosexual family in Jeddah who collects contemporary art and whose African maid mysteriously died in his home. In his country, he is a tyrant at home, enjoying class privileges. Still, his queerness is illegible and illegal. It is suppressed by social norms, legal structures, and familial expectations while the investigation into his maid’s death is covered up. Meanwhile, the very same person, a few times a year, during trips to liberal cities in the U.S. or Europe, navigates a different topos: one where his sexuality is not only legible but celebrated, even capitalized upon. In these art world spaces, his identity is reassembled. He becomes a benevolent queer patron, his philanthropy enhancing his cultural and symbolic capital. Here, queerness is not simply a stable trait but a sheaf-like identity, gathered locally, performed selectively, and stitched into global circuits of prestige. Category theory helps us see this not as contradiction but as relational coherence: identity as a structure that shifts meaningfully depending on the morphisms (social, geographic, legal) that connect one context to another.
Whereas set theory reifies difference as fixed opposition, category theory models difference as structured relation. It invites us to think about how difference connects, how it persists, mutates, or disappears across mappings. This is not an abstract gesture but a political necessity. Under the pressures of algorithmic governance, biometric surveillance, and global migration, identity must be rethought as a diagram of transformation, not a taxonomy of attributes.
Functors and the Politics of Translation
If morphisms describe how individual identities shift across contexts, functors offer a broader framework for how entire systems relate to one another. In category theory, a functor maps between categories in a way that preserves the structure of relationships. It translates not just objects but also the morphisms between them, ensuring the coherence of transformation across domains.
Imagine the “category” of your offline social life. You have relationships like “friend,” “colleague,” and “stranger,” with transformations like meeting a colleague at a work event or encountering a stranger at a conference. Now, consider your online social life. You connect with friends via direct messages, colleagues on LinkedIn, and strangers in comment threads. A functor maps this entire structure from offline to online: it matches each person-type (object) and interaction (morphism) in the offline world to its online counterpart, preserving the structure of how relationships evolve. The functor ensures that if a friend becomes a colleague through an offline work event, the corresponding online transformation (say, a LinkedIn message) reflects this change coherently.
This concept becomes especially powerful when applied to the political, cultural, and institutional infrastructures through which identities are shaped. Consider, for instance, how a single subject (say, a refugee) might be translated across legal, humanitarian, media, and biometric systems. Each system constitutes a distinct category with its own internal logic, yet the individual’s identity must persist, however distorted, across these contexts. The functorial relation is what allows us to track this movement: the mappings that preserve or distort identity in transit.
Functors thus model not sameness but structured variation. They help us understand how identities mutate as they pass from one interpretive regime to another. A trans person may be celebrated within an activist community or in the European Union but illegible in an immigration hearing and criminalized in the United States. A racialized body might be hyper-visible in policing databases and yet invisible in political representation. These are not breakdowns but translations. They are imperfect, noisy, and often asymmetrical.
In this sense, functors make visible the politics of translation: who gets to define coherence, what transformations are allowed, and where rupture is deemed illegible. They enable a diagrammatic politics in which the very act of crossing categories (social, geographic, epistemic) becomes the terrain of struggle.
As the Stanford Encyclopedia of Philosophy notes in discussing category theory, “a category is characterized by its morphisms, and not by its objects.”[5] This insight becomes, in political context, a call to focus on process over position. Rather than fixating on who someone is, we ask: through which systems must they pass, and how does each one rewrite them?
In aesthetic terms, functoriality appears in works that layer systems without seeking a final synthesis: diagrammatic art, procedural texts, or networked performances. Politically, functoriality appears in solidarities that do not rely on shared identity but on the mutual recognition of structural traversal and shared interests and goals. The functor does not unify. It commutes.
To engage functorially is to ask: what transformations must be preserved for an identity to remain politically operative across differences? And what distortions might actually be necessary to survive translation?
Natural Transformations and Differential Identity
While functors map the structure of identities across different categorical regimes (legal, aesthetic, social, computational), they do not fully account for the tensions, slippages, and negotiations that occur when these mappings collide. Natural transformations provide a higher-order concept: they are mappings between functors themselves. They allow us to model how two different systems of translation can relate not in perfect symmetry, but through structured deviation.
Imagine a category of Global South countries: Nigeria, Peru, and Indonesia. The U.S. and China each enact influence over these countries through structured programs: IMF loans, trade deals, and military aid for the U.S.; BRI projects, Chinese trade deals, and security pacts for China.
Each country is mapped to one of these strategies: this mapping is a functor. One functor goes from the Global South to the U.S. geopolitical toolkit and another to China’s. These functors preserve structural relationships: for example, economic leverage aligns with infrastructure investment, and military aid aligns with security agreements.
However, the mappings are not identical, and this is where natural transformations come in. The dashed arrows between U.S. and Chinese strategies represent structured deviations: they model the tensions and negotiations between competing systems of influence. A country like Nigeria might receive an IMF loan (via the U.S. functor) while also signing a BRI deal (via the China functor). The natural transformation captures how these pathways relate, conflict, or translate into one another.
In categorical terms, natural transformations let us model not just identity shifts but systemic friction, where influence strategies diverge, overlap, or transform in response to one another.
In the realm of identity, natural transformations name the interface between systems that translate differently: how medical, legal, and familial systems each construct “trans identity” in ways that are structurally consistent within themselves yet mutually incompatible. A natural transformation does not resolve this incompatibility but formalizes it. It reveals how translation can proceed even when structural logics diverge.
To speak of differential identity, then, is to foreground these variational processes. Identity is not simply what moves through systems but the difference that emerges between systems. It is incommensurability rendered legible not as an error, but as structure. Differential identity is what gets preserved and what gets distorted across all mappings. It is the internal logic of the misfit.
This is not only an abstract concern. Consider how a Palestinian citizen of Israel, an undocumented trans person in the U.S., a Uyghur activist in exile, or a Ukrainian caught in the oppression of Russian occupation of east Ukraine must navigate legal systems that rewrite their identities with each crossing. The lived experience of contradiction is not a failure of coherence but a diagram of pressure, survival, and transformation. Differential identity is a complex political geometry.
Reza Negarestani’s concept of “inhumanism,”[6] Fred Moten’s notion of fugitivity,[7] and Édouard Glissant’s right to opacity[8] all gesture toward this terrain. These are not fixed identities but modalities of movement. They are diagonal traversals across frames that never fully align. Natural transformations describe how such movements remain structurally legible even when they refuse the stabilizing pull of identity as sameness.
This opens up the possibility for a new kind of solidarity, one not built on identification but on the recognition of structurally shared misalignments. To see someone else’s differential identity is not to assimilate them into your frame but to acknowledge that both of you are being translated, warped, and suspended by systems neither of you fully control.
Functional Aesthetics and Diagrammatic Resistance
If functors reveal how identities move through and between systems, and natural transformations reveal how such systems themselves relate through structured distortions, then functional aesthetics names a domain where these processes are felt, performed, and rendered perceptible. In a world saturated by images, data, and coded gestures, aesthetics becomes not a realm of decoration or beauty but a domain of operational abstraction. It is where identity is not merely represented but diagrammed. Identity is made to function as a system.
A functional aesthetic is one that privileges behavior over resemblance, procedure over picture. It maps how identities circulate, mutate, and stabilize through different infrastructures. Think of Thomas Bayrle’s use of repetitive units or Trevor Paglen’s training datasets. These are works that do not illustrate identity but rather instantiate the logic of pattern recognition, surveillance, or propagation. Their aesthetic force emerges not from symbolic meaning but from infrastructural participation: these are not about identity but about how identity is processed.
This diagrammatic logic is not new. Geometric abstraction from Iran and the Arab world, West African textile systems, and Indigenous cartographic traditions have long encoded knowledge in non-representational forms. In these traditions, identity is woven, repeated, performed. It is structured through rhythm and recursion rather than fixed depiction. A Kente cloth or a Mixtec codex does not tell you what something is, but how it relates, how it moves, how it inherits and transforms.
Diagrammatic resistance arises when these modes are turned against capture. It is the refusal to be flattened into data while still engaging the systems that demand legibility. It is the strategic deployment of opacity, modulation, recursion, and abstraction not as a retreat but as intervention. Tactical illegibility. Glissant’s right to opacity becomes, here, a politics of interface: not transparency, but translated depth.
In the hands of contemporary artists, designers, and coders, this resistance appears in generative algorithms, feedback loops, speculative diagrams, and encrypted social gestures. The diagram becomes a weapon not because it explains but because it reorganizes perception. It redistributes what counts as intelligible. It stretches coherence just far enough to preserve relation while denying extraction.
Functional aesthetics and diagrammatic resistance do not oppose politics; they enact it on another register. They make visible the infrastructure of identity while refusing to be reduced to it. They are acts of form that rewire function.
Toward a Diagrammatic Politics
To conclude, if identity today is marked by exhaustion, capture, and operational overexposure, diagrammatic politics offers a pathway beyond both essentialism and post-political relativism. The diagram (borrowed from category theory but extended into aesthetic, cultural, and political life) is not simply a tool for mapping what is. It is an architecture for composing new modes of relation.
A diagrammatic politics rejects the fantasy of pure visibility and stable classification. It proceeds by tracking movements, coordinating transformations, and organizing transversal solidarities. It does not seek to anchor the subject but to facilitate its freedom and movement. In this sense, the diagram is not just a representation of change but a performative infrastructure through which change becomes legible and actionable.
Such a politics foregrounds three interwoven strategies: first, a refusal of reduction (identity as irreducible to any single frame); second, a fidelity to structure (not as rigid form but as a space of constraint and experimentation); and third, a commitment to transformation (identity as emergent through contingent, partial, and situated mappings).
This is not a call to abandon identity politics but to reformat it. What we need is not less identity but better diagrams: maps that do not fix us in place but chart the vectors of our becoming. A diagrammatic politics allows us to think across systems without collapsing their differences, to move without erasure, and to articulate solidarity not through sameness but through shared misalignment.
As such, the political subject of the diagram is not the self-possessed individual but the relational composite: a morphism among morphisms, a structure among structures. Identity becomes a field of operations. It is unstable, situated, recursive, and potent.
To diagram is to resist both the static identity and the nihilistic void. It is to trace structure in flux. It is to hold form and transformation together not as a contradiction, but as a condition.
- This paper was presented on May 24, 2025, at the Identity Crisis Network Conference in Zagreb.
Notes
[1] Samuel Eilenberg and Saunders Mac Lane, “General Theory of Natural Equivalences,” Transactions of the American Mathematical Society 58, no. 2 (1945): 231-294. This foundational paper introduced the concepts of categories, functors, and natural transformations.
[2] Fernando Zalamea, Synthetic Philosophy of Contemporary Mathematics, trans. Zachary Luke Fraser (London: Urbanomic/Sequence Press, 2012). Originally published in Spanish in 2009, this work provides a comprehensive philosophical analysis of contemporary mathematics, with extensive discussion of Grothendieck’s innovations in category theory.
[3] Alexander Grothendieck, “Esquisse d’un Programme” (Sketch of a Programme), written in 1984 as a research proposal for CNRS, published in Geometric Galois Actions: Volume 1, Around Grothendieck’s Esquisse d’un Programme, eds. Leila Schneps and Pierre Lochak, London Mathematical Society Lecture Note Series 242 (Cambridge: Cambridge University Press, 1997), 5-48.
[4] Jean-Pierre Serre, “Motifs,” in Journées arithmétiques de Luminy 1989, ed. G. Lachaud, Astérisque 198-200 (1991): 335. In French: “comme Grothendieck nous l’a appris, les objets d’une catégorie ne jouent pas un grand rôle, ce sont les morphismes qui sont essentiels” (translation: “as Grothendieck taught us, the objects of a category do not play a large role; it is the morphisms that are essential”).
[5] Jean-Pierre Marquis, “Category Theory,” Stanford Encyclopedia of Philosophy, first published December 6, 1996; substantively revised September 18, 2019, https://plato.stanford.edu/entries/category-theory/.
[6] Reza Negarestani develops the concept of “inhumanism” across several works, most notably in “The Labor of the Inhuman, Part I: Human,” e-flux journal #52 (February 2014), https://www.e-flux.com/journal/52/59920/the-labor-of-the-inhuman-part-i-human/, and “The Labor of the Inhuman, Part II: The Inhuman,” e-flux journal #53 (March 2014). His rationalist inhumanism is further elaborated in Intelligence and Spirit (London: Urbanomic & Sequence Press, 2018). Negarestani describes inhumanism as “the extended practical elaboration of humanism; it is born out of a diligent commitment to the project of enlightened humanism. A universal wave that erases the self-portrait of man drawn in sand, inhumanism is a vector of revision.”
[7] Fred Moten and Stefano Harney, The Undercommons: Fugitive Planning & Black Study (New York: Minor Compositions/Autonomedia, 2013). Moten develops the concept of fugitivity as a mode of existence that refuses capture and operates outside the bounds of institutional control, exploring “fugitive planning” and “Black study” as modes of resistance rooted in the Black radical tradition.
[8] Édouard Glissant, Poetics of Relation, trans. Betsy Wing (Ann Arbor: University of Michigan Press, 1997), 189-194. Originally published as Poétique de la Relation (Paris: Gallimard, 1990). Glissant writes: “We clamor for the right to opacity for everyone” (“Nous réclamons pour tous le droit à l’opacité”). The concept refers to the right of individuals and cultures to remain irreducible to complete understanding or transparency, resisting colonial demands for total legibility. Glissant first articulated this demand in Caribbean Discourse (1981) and continued developing it throughout his later works.
* This paper was presented on May 24, 2025, at the Identity Crisis Network Conference in Zagreb.